An introduction to Hopf algebras

An Introduction to Hopf Algebras

Robert G. Underwood

An Introduction to Hopf Algebras

123

Robert G. Underwood Department of Mathematics Auburn University Montgomery, AL 36124 USA [email protected]

ISBN 978-0-387-72765-3 e-ISBN 978-0-387-72766-0 DOI 10.1007/978-0-387-72766-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011933563 Mathematics Subject Classification (2010): 13AXX, 16TXX, 20AXX © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my wife, Rebecca

Preface

The purpose of this book is to provide an introduction to Hopf algebras. This book differs from other texts in that Hopf algebras are developed from notions of topological spaces, sheaves, and representable functors. This approach has certain pedagogical advantages, the foremost being that algebraic geometry and category theory provide a smooth transition from modern algebra to Hopf algebras. For example, the motivation for the definition of an exact sequence of Hopf algebras is best established by first defining exactness for a sequence of representable group functors. Hopf algebras are attributed to the German mathematician Heinz Hopf (1894–1971). The study of Hopf algebras spans many fields in mathematics, including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups. In this work, we focus on applications of Hopf algebras to algebraic number theory and Galois module theory. By the end of the book, readers will be familiar with established results in the field and should be poised to phrase research questions of their own. An effort has been made to make this book as self-contained as possible. That said, readers should have an understanding of the material on groups, rings, and fields normally covered in a basic course in modern algebra. Also, most of the groups given here are Abelian. All of the rings are non-zero commutative rings with unity, and we only consider commutative algebras. This work contains 12 chapters. Each chapter contains an exercise set with questions ranging in level of difficulty from elementary to advanced. In the last chapter, we list some open problems and research questions. In Chapter 1, we begin by introducing Spec A, the spectrum of a commutative ring with unity A. Included is a section on the role played by nilpotent elements as the zero functions on Spec A. In Chapter 2, we endow Spec A with the Zariski topology and study its properties as a topological space. We next introduce presheaves and sheaves on Spec A. The structure sheaf on Spec A is then generalized to the functor HomR-alg .A; /, represented by the R-algebra A. In Chapter 3, we endow HomR-alg .A; / with the structure of a group functor, which leads in Chapter 4 to the definition of an R-Hopf algebra H . In this chapter, we prove in detail vii

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the fundamental result of R. Larson and M. Sweedler that states that if H is a cocommutative Hopf algebra of finite rank over a Dedekind domain R R R, then the linear dual H  is isomorphic to H ˝R H  as H -modules, where H  is the ideal of integrals of H  . We specialize to Hopf orders in group rings in Chapters 5 through 9. In Chapter 5, we review R. Larson’s construction of Hopf orders in KG, where G is an arbitrary finite Abelian group, and in Chapter 6 we employ formal groups to construct a collection of R-Hopf orders in KCpn , where Cpn denotes the cyclic group of order p n . The material in this chapter is joint work with Lindsay Childs of the University at Albany, SUNY. In Chapter 7, we review the classification of Hopf orders in KCp due to J. Tate and F. Oort, and in Chapter 8 we employ C. Greither’s cohomological argument to give a complete classification of Hopf orders in KCp2 in the case where K is a finite extension of Qp containing a primitive p 2 nd root of unity. In Chapter 9, we review several constructions of Hopf orders in KCp3 (also joint work with L. Childs). We next turn to some important applications of Hopf algebras. In Chapter 10, we use results of Lindsay Childs and Nigel Byott to describe the ring of integers in an algebraic number field as a Hopf-Galois module. In Chapter 11, we give a topological proof of the finiteness of the class group C.R/, following J. W. S. Cassels. We then generalize the class group to Hopf orders H and give A. Fr¨ohlich’s hom-description of the class group C.H /. Next, we investigate the structure of the Hopf-Swan subgroup of C.H /. In Chapter 12, we discuss some open questions and research problems. This book grew out of lectures given during a year-long course in modern algebra at Auburn University, Montgomery Campus, over the period 2006–2007. I would like to thank Professors Lindsay Childs, Jorg Feldvoss, James Carter, Tim Kohl, and the late Bettina Richmond, who kindly agreed to read a draft of the book. I am especially grateful to Lindsay Childs, who introduced me to Hopf algebras and whose body of work has informed and influenced my study of them. I would also like to thank former and current students who read and commented on early drafts of the class notes. I wish to thank the Department of Mathematics at Auburn University, Montgomery, for its support during my sabbatical year, 2008–2009, and the Department of Mathematics at Florida State University for its hospitality and support during my sabbatical year while I worked on this book. In particular, I thank Professor Warren Nichols for our many discussions about coalgebras, bialgebras, Hopf algebras, and data mining. I wish to thank my editors at Springer, Ann Kostant and Elizabeth Loew, for their patience with my earlier drafts and their support and guidance throughout this project. Finally, I thank my wife, Rebecca Brower, my son, Andre, and my parents, especially my father, the late Kenneth L. Underwood (1933–2010), who have been a great inspiration to me and who have given me the strength to persevere and complete this work.

Preface

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Figures 1.1 and 1.2 originally appeared as Figures 13 and 14, respectively, in c Springer-Verlag, 1974, and are Basic Algebraic Geometry by I. R. Shafarevich,  used with kind permission of Springer Science and Business Media. February 2011

Robert G. Underwood

Contents

1

The Spectrum of a Ring .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Introduction to the Spectrum . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 The Associated Map of Spectra . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Nilpotent Elements .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 3 8 11

2

The Zariski Topology on the Spectrum . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Some Topology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Basis for a Topological Space . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Representable Functors . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13 13 17 23 27 32

3

Representable Group Functors . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction to Representable Group Functors .. . . . . . . . . . . . . . . . . . . . 3.2 Homomorphisms of R-Group Schemes .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Short Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

35 35 41 47 51 53

4

Hopf Algebras .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction to Hopf Algebras .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Dedekind Domains .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Hopf Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Hopf Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

55 55 65 70 82 93

5

Valuations and Larson Orders . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 5.1 Valuations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 5.2 Group Valuations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 5.3 Larson Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 104 5.4 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 113

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6

Formal Group Hopf Orders .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Formal Groups.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Formal Group Hopf Orders . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

115 115 121 128

7

Hopf Orders in KCp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129 7.1 Classification of Hopf Orders in KCp . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129 7.2 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139

8

Hopf Orders in KCp2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 The Valuation Condition .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Some Cohomology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Greither Orders .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Hopf Orders in KC4 , KC9 . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

141 141 148 155 171 179

9

Hopf Orders in KCp3 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Duality Hopf Orders in KCp3 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Circulant Matrices and Hopf Orders in KCp3 . .. . . . . . . . . . . . . . . . . . . . 9.3 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

181 181 185 194

10 Hopf Orders and Galois Module Theory . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Some Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Galois Extensions of Rings . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Hopf-Galois Extensions of a Local Ring . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5 The Normal Basis Theorem.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

195 195 202 213 220 226 230

11 The Class Group of a Hopf Order . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 The Class Group of a Number Field. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 The Class Group of a Hopf Order . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 The Hopf-Swan Subgroup . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

233 233 240 246 259

12 Open Questions and Research Problems.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 The Spectrum of a Ring .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3 Valuations and Larson Orders . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.4 Hopf Orders in KCp2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.5 Hopf Orders in KCp3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.6 Hopf Orders and Galois Module Theory .. . . . . . .. . . . . . . . . . . . . . . . . . . . 12.7 The Class Group of a Hopf Order . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

261 261 261 262 262 263 264 265

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 267 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 271

Notation

Z Z>0 Q Q R R>0 R C 1A A m AB AB H G H
ring of integers positive integers field of rational numbers non-zero rationals field of real numbers positive real numbers non-zero reals complex numbers unity in the ring A structure map of the R-algebra A primitive mth root of unity A is a subset of B A is a proper subset of B H is a subgroup of G H is a proper subgroup of G H is a normal subgroup of G order of the finite group G image of a under canonical surjection A ! A=B group of the nth roots of unity cyclic group of order m character group of G index of H in G degree of L over K localization of R at prime ideal P completion of R at prime ideal P uniformizing parameter group of units in R of the form 1 C R j finite field with p elements ring of p-adic integers field of p-adic rationals normalized absolute value on K xiii

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K disc.S=R/ Spec A RŒF  Gm Ga n HomR-alg .A; B/ H H H

H HD H h; i  A./ „.H / H.i /, H.i; j / F .x; y/ E.G; Cm / A.i; j; u/ A.i; j; k; u; v; w/ Gal.L=K/ GE.H; R/ S GE.H; R/ VK JK IK F .R/ C.R/ C.H / R.H / T .H /

Notation

completion of K with respect to the j j -topology discriminant of S over R spectrum of the ring A representing algebra of the representable functor F multiplicative group scheme additive group scheme multiplicative group of the nth roots of unity R-algebra homomorphisms from A into B Hopf algebra, Hopf order comultiplication map of H counit map of H coinverse of H dual module of H linear dual of H duality map H   H ! R order-bounded group valuation Larson order given by  order-bounded group valuation determined by H Larson order in KCp , KCp2 , respectively n-dimensional degree 2 formal group equivalence classes of extensions of G by Cm Greither order in KCp2 duality Hopf order in KCp3 Galois group of L over K Galois H -extensions of R semilocal Galois H -extensions of R adele ring of K group of ideles of K ideal group of K fractional ideals of K class group of R class group of the Hopf order H realizable classes in C.H / Hopf-Swan subgroup of C.H /

Chapter 1

The Spectrum of a Ring

1.1 Introduction to the Spectrum Throughout this chapter, by “ring” we mean a non-zero commutative ring with unity. Our first proposition employs Zorn’s Lemma [Rot02, Appendix]. Let S be a non-empty set, and let  be a relation on S . We say that S is partially ordered if  is reflexive, antisymmetric, and transitive. A subset T of a partially ordered set S is a chain if for all x; y 2 T either x  y or y  x. Zorn’s Lemma. Let X be a non-empty partially ordered set in which each chain has an upper bound. Then X has a maximal element. Proposition 1.1.1. Let A be a ring. Then every proper ideal of A is contained in a prime ideal. Proof. Let J be a proper ideal of A, and let P denote the collection of all proper ideals of A that contain J . Since J 2 P, P is a non-empty set that is partially ordered under set inclusion. Let C be a chain in P. Then the ideal [I 2C I is an upper bound for C. Thus, by Zorn’s Lemma, P contains a maximal element M . We show that M is a maximal ideal by showing that A=M is a field. Since A is a commutative ring with unity, then so is A=M . Let b be an element of A with .b/ C M 6D M . Since M  .b/ C M , .b/ C M cannot be in P, and so M  .b/ C M D A. Since 1 2 A, there exist elements r 2 A and m 2 M such that rb C m D 1 and hence rb D 1  m. Now .r C M /.b C M / D rb C M D .1  m/ C M D .1 C M / C .m C M / D 1 C M; and thus b C M is a unit of A=M . Consequently, A=M is a field such that M is a maximal ideal of A. Since every maximal ideal is a prime ideal, the result follows. R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 1, © Springer Science+Business Media, LLC 2011

1

2

1 The Spectrum of a Ring

Definition 1.1.1. Let A be a ring. The collection of prime ideals of A is the spectrum of A and will be denoted by Spec A. The elements of the spectrum are the points of Spec A. Proposition 1.1.1 shows that Spec A is non-empty. When the ideal .0/ is a prime ideal of A, and hence is a point of Spec A, we write ! D .0/. For example, Spec Z D f.2/; .3/; .5/; .7/; .11/; : : : ; !g. Proposition 1.1.2. Let F be a field. Then Spec F D f!g. Proof. The only ideals of F are f0g and F , and of these only f0g is prime.

t u

Let A be a ring and let x 2 Spec A. Then S D Anx is a multiplicative set and we can form the localization of A at x defined as S 1 A D fa=s W a 2 A; s 2 S g: We denote the localization of A at x by Ax . In the case where A D Z and p is a prime of Z, we have the following. Proposition 1.1.3. Let p be a prime of Z. Then Spec Z.p/ D f!; .p/g. Proof. Since Q is an integral domain and Z.p/  Q, Z.p/ is an integral domain. Therefore, ! 2 Spec Z.p/ . Moreover, Z.p/ is a local ring with unique maximal ideal .p/, and so .p/ 2 Spec Z.p/ . We show that ! and .p/ are the only points of Spec Z.p/ . Let I be a proper ideal of Z.p/ other than ! or .p/. By Proposition 1.1.1, I is contained in a prime ideal; in fact, it is contained in a maximal ideal and consequently I  .p/. Now I \ Z  .p/ \ Z D pZ, and so I \ Z D p l Z with l > 1. Thus, in Z.p/ =I , l p D 0, and so Z.p/ =I is not an integral domain and hence I is not a prime ideal. It follows that ! and .p/ are the only points in Spec Z.p/ . t u More generally, let A be a ring and let x 2 Spec A. Then Spec Ax D fyAx W y 2 Spec A; y  xg: An element a 2 A is nilpotent if an D 0 for some integer n > 0. If f is a non-nilpotent element of A, then S D f1; f; f 2 ; f 3 : : : g is a multiplicative set that determines the localization S 1 A. Proposition 1.1.4. Let f be a non-nilpotent element of A with S D f1; f; f 2 ; : : : g, and let Af denote the localization S 1 A. Then Spec Af D fxAf W x 2 Spec A; f 62 xg: Proof. Exercise.

t u

For example, Spec Zf , f D 6, consists of f.5/; .7/; .11/; : : : ; !g. To compute the spectrum of a quotient ring, we need the following proposition.

1.2 The Associated Map of Spectra

3

Proposition 1.1.5. Let  W A ! B be a ring homomorphism. Suppose x is a prime ideal of B. Then  1 .x/ is a prime ideal of A. Proof. We first show that  1 .x/ is an ideal of A. Since  is an additive group homomorphism,  1 .x/ is an additive subgroup of A. Let a 2 A and let b 2  1 .x/. Then .ab/ D .a/.b/ D .a/c, where c 2 x. Thus .ab/ 2 x, so that ab 2  1 .x/. We next show that  1 .x/ is prime. Suppose ab 2  1 .x/. Then .ab/ D .a/.b/ 2 x, so that either .a/ 2 x or .b/ 2 x. This shows that either a 2  1 .x/ or b 2  1 .x/, and so  1 .x/ is prime. t u Proposition 1.1.6. Let I be an ideal of A. Then Spec .A=I / is in a one-to-one correspondence with the collection of prime ideals x of A for which I  x; that is, Spec .A=I / D fx W x 2 Spec A; I  xg; where x denotes the image of x under the canonical surjection A ! A=I . Proof. Let x be a prime ideal of A containing I . Then x is an additive subgroup of A=I . For a 2 A=I , a  x D ax  x, and so x is an ideal of A=I . We show that x is prime. Suppose f  g 2 x. Then fg 2 x; that is, fg D s for some s 2 x. Hence, fg  s D 0, and so fg 2 s C I  x. Thus either f 2 x or g 2 x, and so either f 2 x or g 2 x. Thus x is in Spec A=I . Now suppose x is a prime ideal of A=I . Then, by Proposition 1.1.5, x is a prime ideal of A. t u For example, let A D Z, I D .6/. Then Spec Z=.6/ D f.2/; .3/g, where .2/ D f0; 2; 4g and .3/ D f0; 3g. One may wonder why we don’t consider the spectrum of all maximal ideals of A instead of just prime ideals. Proposition 1.1.5 helps explain why: Proposition 1.1.5 fails if we replace “prime ideal” with “maximal ideal.” For example, the inclusion of { W Z ! Q is so that !  Q is maximal while { 1 .!/ D !  Z is not.

1.2 The Associated Map of Spectra In this section, we introduce the basic map between two spectra. Let  W A ! B be a ring homomorphism with .1A / D 1B , and let x 2 Spec B. By Proposition 1.1.5,  1 .x/ 2 Spec A, and thus there exists a map Spec B ! Spec A defined by x 7!  1 .x/ for x 2 Spec B. We call this map the associated map of spectra and denote it by a .

4

1 The Spectrum of a Ring

Example 1.2.1. Let .p/ be a non-zero prime ideal of Z, and consider the inclusion  W Z ! Z.p/ given by n 7! n=1. By Proposition 1.1.3, Spec Z.p/ D f!; .p/g. The associated map a  W Spec Z.p/ ! Z is given as a .!/ D .!/; a ..p// D .p/. Example 1.2.2. Let A D Z, and consider the canonical surjection | W Z ! Z=.15/. Then a | W Spec Z=.15/ ! Spec Z is given as .3/ 7! .3/; .5/ 7! .5/. Let ZŒi denote the Gaussian integers. We know that the integers can be embedded into the Gaussian integers by the map { W Z ! ZŒi defined as {.a/ D a, a 2 Z. We compute the spectrum of ZŒi and the associated map a { W Spec ZŒi ! Spec Z. Since ZŒi is an integral domain, ! 2 Spec ZŒi, and since { 1 .!/ D !, a {.!/ D !. Candidates for the other elements of Spec ZŒi include the principal ideals .2/; .3/; .5/; : : : of ZŒi. However, we must check whether these ideals are prime in ZŒi. For example, one has 2 D .1 C i/.1  i/, and so .2/ is not a prime ideal of ZŒi. Moreover, .1 C i/ D .1  i/ since i.1 C i/ D .1  i/. We claim that .1 C i/ is prime. In the quotient ZŒi=.1 C i/, i D 1; thus ZŒi= .1 C i/ Š Z, which is an integral domain, and so .1 C i/ is prime. Moreover, { 1 .1 C i/ D .2/, and so a {.1 C i/ D .2/. But what about the ideals generated by the odd primes of Z? The following proposition, known as Fermat’s Two-Squares Theorem, tells us precisely when .p/, p > 2, factors in ZŒi. Proposition 1.2.1. Suppose p > 2. Then p D .a C bi/.a  bi/ for some integers a; b if and only if p  1 .mod 4/. Proof. We first suppose that p factors as .a C bi/.a  bi/ and consider cases determined by whether a and b are even or odd. Case 1. a and b are both even or a and b are both odd. In this case, p is even, which is impossible. Case 2. a is even and b is odd (or vice versa). Without loss of generality, we suppose that a D 2k, b D 2l C 1, for some integers k; l. Then p D .2k/2 C .2l C 1/2 D 4k 2 C 4l 2 C 4l C 1  1 .mod 4/: This proves the “only if” part of the theorem. For the converse, we suppose p  1 .mod 4/ and show that there exist integers a; b, such that p D a2 C b 2 . Let m be such that p  1 D 4m. Then m D .p  1/=4. Since Z=.p/ is a finite field, its multiplicative group of units, U.Z=.p//, is cyclic and isomorphic to Z=.p  1/. Let r be a generator for U.Z=.p//. Since the order of r in Z=.p/ is p  1, r 2m  1 .mod p/. Put s D r m . Then s 2 C 1  0 .mod p/, so that s 2 C 1 D .s C i/.s  i/ D pn for some n 2 Z. Now, if .p/ is a prime ideal in ZŒi, then either s C i 2 .p/ or s  i 2 .p/, or, in other words, either p divides s C i or p divides s  i. We may assume without losing generality that p.c C d i/ D s C i for integers c; d . But this

1.2 The Associated Map of Spectra

5

says that pd D 1, which is a contradiction. So .p/ is not prime and thus, since ZŒi is a PID, p is reducible. Hence p D .a C bi/.c C d i/ for some integers a; b; c; d . But this implies that p D .a  bi/.c  d i/, and so p 2 D .a2 C b 2 /.c 2 C d 2 /: Now, since p is prime, p D a2 C b 2 , and hence p D .a C bi/.a  bi/.

t u

Now we know precisely when .p/ is prime in ZŒi. Proposition 1.2.2. Suppose p > 2. The ideal .p/ is prime in ZŒi if and only if p 6 1 .mod 4/. Proof. By Proposition 1.2.1, p is a reducible element of ZŒi if and only if p  1 .mod 4/. Thus .p/ is a prime ideal in ZŒi if and only if p 6 1 .mod 4/. u t When p, p > 2, does factor, it turns out that the ideals .a C bi/ and .a  bi/ are distinct prime ideals of ZŒi. In fact, these ideals are maximal. In what follows, we put Fp D Z=.p/. Proposition 1.2.3. Suppose p > 2, p D .a Cbi/.a bi/. Then .a ˙bi/ are distinct maximal ideals of ZŒi. Proof. Put I D .a C bi/. We show that ZŒi=I is a field. We have x C yi  z C yi .mod I / if and only if x  z .mod p/. Likewise, x C yi  x C wi .mod I / if and only if y  w .mod p/. Thus ZŒi=I Š Fp Œi=J , where J D .a C bi/. Here, a; b are the images of a; b under the canonical surjection Z ! Fp . Now b is a unit in Fp , and hence J D .c C i/ for some c 2 Fp . Thus, in the quotient Fp Œi=J , i  c, and so Fp Œi=J Š Fp . Thus, ZŒi=I Š Fp , a field. A similar argument shows that ZŒi=.a  bi/ Š Fp . We leave it as an exercise to show that .a C bi/ 6D .a  bi/. t u If .p/ is prime in ZŒi, then ZŒi=.p/ is an integral domain with p 2 elements, and hence ZŒi=.p/ is a field. In fact, every finite field has q k elements for some prime q and integer k  1 [La84, Chapter 7, 5]. We now have Spec ZŒi computed: Spec ZŒi D f.1 C i/; .3/; .2 C i/; .2  i/; .7/; .11/; : : : ; !g: The spectral diagram that illustrates the associated map a  W Spec ZŒi ! Spec Z appears as Fig. 1.1 below. Let C2 D hi denote the cyclic group of order 2. Another important computation of the spectral diagram involves the group ring ZC2 D fa C b W a; b 2 Zg: There is a natural inclusion { W Z ! ZC2 , defined as a 7! a1 C 0. As we did with the Gaussian integers, we compute the associated map a { W Spec ZC2 ! Spec Z.

6

1 The Spectrum of a Ring .2 C i/ .1 C i/

.3/

r

Spec ZŒi

r

.3 C 2i/ .7/

r

r

.11/

r

r

r

r

.2  i/

.3  2i/

+

+

.2/

.3/

.5/

.7/

r

r

r

r

Spec Z

!

r

.11/

.13/

r

r

!

r

Fig. 1.1 The map of spectra a { W Spec ZŒi  ! Spec Z

Proposition 1.2.4. Suppose p > 2. Then  pD

pC1 p1 C  2 2



 pC1 p1   : 2 2

Proof. Observe that  pD

pC1 2

2

 

p1 2

2

 D

pC1 2

2

and factor as the difference of two squares.

 

p1  2

2 ; t u

Let us try to recover this factorization using the method of Proposition 1.2.1. Since p is odd, m D .p  1/=2 is an integer. Let r be a generator for the multiplicative group of Fp , and put s D r m . Then s 2  1  0 .mod p/, so that s 2  1 D .s C /.s  / D pn for some n 2 Z. Now, if .p/ is a prime ideal in ZC2 , then p divides either s C  or s  . We may assume without losing generality that p.c C d / D s C  for integers c; d . But this says that pd D 1, which is a contradiction. So .p/ is not a prime ideal. But does p factor in ZC2 ? The answer is “yes,” but not because .p/ fails to be prime in ZC2 (ZC2 is not a PID, so we do not automatically know that p is reducible). We seek integers a; b; c; d such that p D .a C b/.c C d /: If such integers exist, then p D ac C bd and 0 D ad C bc, so that pa D a 2 c C abd and 0 D abd Cb 2 c. Thus b 2 c D abd , which yields pa D a2 cb 2 c. So let c D a.

1.2 The Associated Map of Spectra

7

Then p D a2  b 2 D .a C b/.a  b/. Now, since p is prime in Z, we have either p D a C b, p D a  b, p D a  b, or p D a C b. Without loss of generality, we can assume that p D a C b. Then 1 D a  b, so that 2a D p C 1, and so a D c D pC1 and b D p1 , which yields p D . pC1 C p1 /. pC1  p1 /. 2  2  2 2 2   2 pC1 p1 pC1 p1 C  Put I D 2 C 2  and I D 2  2  . Proposition 1.2.5. For p > 2, the ideals I C and I  are distinct prime ideals of ZC2 . Proof. We first show that the ideals I ˙ are maximal. Arguing as in Proposition 1.2.3, one concludes that ZC2 =I C Š Fp C2 =I C , where IC D

pC1 p1 C  2 2

! D .1 C /:

Thus ZC2 =I C Š Fp C2 =.1 C / Š Fp , which is a field. Consequently, I ˙ are maximal ideals. Now suppose I C D I  . Then pC1 p1 pC1 p1 C C   D p C 1 2 I C; 2 2 2 2 which is impossible since p 2 I C and .p; p C 1/ D 1.

t u

Next, we consider the prime ideal .2/  Z. Since .1 C /  0 .mod 2/, the ideal .2/  ZC2 is not prime. It is irreducible, however: there is no factorization of 2 in ZC2 . So, which prime ideals x 2 Spec ZC2 satisfy 1 .x/ D .2/? Any such prime ideal x contains .2/, and since 2 is irreducible, x is an ideal of the form .2; a C b/ for some a C b 2 ZC2 . Since F2 C2 =.a C b/ is an integral domain if and only if a D b D 1, x D .2; 1 C /. Finally, the prime ideal ! of Z factors into two distinct prime ideals .1 C / and .1  / of ZC2 . We conclude that 2

Spec ZC2 D f.2; 1 C /; .2  /; .2 C /; .3  2/; .3 C 2/; : : : ; .1 C /; .1  /g; and the spectral diagram appears as Fig. 1.2 below. For another example, let C3 denote the cyclic group of order 3, generated by . Using the ideas employed in the computation of Spec ZC2 , one can compute Spec ZC3 as well as the map of spectra a { W Spec ZC3 ! Spec Z induced by the embedding { W Z ! ZC3 . Note that ZC3 has zero divisors: .  1/. 2 C  C 1/ D 0. To compute the preimage of .2/  Z in Spec ZC3 , observe that x 2 C x C 1 is irreducible over F2 ; consequently, .2;  2 C  C 1/ is a maximal ideal of ZC3 . The complete diagram appears as Fig. 1.3 below.

8

1 The Spectrum of a Ring

.2   /

r

.2; 1 C  /

.3  2 /

r

.

pC1 2

.

pC1 2

r



p1 / 2

C

p1 / 2

.1 C  /

r

r

Spec ZC2

r .2 C  /

r

r

.3 C 2 /

+

Spec Z

r .1   /

+

.2/

.3/

.5/

.p/

r

r

r

r

!

r

Fig. 1.2 The map of spectra a { W Spec ZC2 ! Spec Z Spec ZC3

.13;  C 4/

.7; 2   /

r

.5;  2 C  C 1/

r

.2;  2 C  C 1/

r

.3;   1/

r

r

.7;   1/

r

.5;   1/

r

r .11;   1/

+ Spec Z

r

.11;  C  C 1/

.13;   3/

.7;  C 3/

r

. 2 C  C 1/

2

r

r

.2;   1/

r

.  1/

r

r

.13;   1/

+

.2/

.3/

.5/

.7/

.11/

.13/

r

r

r

r

r

r

!

r

Fig. 1.3 The map of spectra a { W Spec ZC3 ! Spec Z

1.3 Nilpotent Elements Let A be a ring. Recall that an element f 2 A is nilpotent if there exists a positive integer n for which f n D 0.

1.3 Nilpotent Elements

9

Proposition 1.3.1. The collection of nilpotent elements forms an ideal of A. Proof. We first show that the set S of nilpotent elements is a subgroup. Let a; b 2 S with an D 0, b m D 0, for positive integers m; n. Then .a C b/mCn1 D

mCn1 X i D0

! m C n  1 mCn1i i a b: i

If m C n  1  i  n, then the term in the sum on the right-hand side is 0. Otherwise, if n > mCn1i , then i > m1 and so the sum on the right is 0. Thus aCb 2 S . Clearly, 0 2 S . Moreover, .a/n D an D 0 if n is even, and .a/n D an D 0 if n is odd, so that a 2 S if a 2 S . Thus S is a subgroup of A. Let r 2 A, a 2 S . Then .ra/n D r n an D 0, and so ra 2 S . Thus S is an ideal of A. t u The ideal of nilpotent elements is called the nilradical of A and is denoted by nil.A/. The nilpotent elements play an important role in the study of Spec A. Let x 2 Spec A. Then A=x is an integral domain and as such has a field of fractions Frac.A=x/. There is a natural inclusion {x W A=x ! Frac.A=x/, depending on x, and the composition of this map with the canonical surjection |x W A ! A=x yields the ring homomorphism `x W A ! Frac.A=x/; `x D {x |x : Let f 2 A. Then f determines a function on Spec A, f W Spec A !

[

Frac.A=x/;

x2Spec A

where f .x/ D `x .f / for x 2 Spec A. In this manner, A can be viewed as a collection of functions on Spec A. Example 1.3.1. Let A D Z, and let f D 12. Then 12 determines a function on Spec Z, 1 0 [ [ 12 W Spec Z ! Frac.Z=x/ D @ Fp A [ Q; x2Spec Z

p2Z

with 12..2// D 0, 12..3// D 0, 12..5// D 2, 12..7// D 5, 12..11// D 1, and 12.x/ D 12 for all other points x 2 Spec Z. In Example 1.3.1, the first and second zeros are taken to be equal. This is technically incorrect, however, since the first zero is the residue class of 0 in F2 , while the second 0 is the residue class in F3 . But for our purposes we shall assume they are equal.

10

1 The Spectrum of a Ring

In the ring A, the element 0 2 A is a zero function on Spec A since 0.x/ D 0 for all x 2 Spec A. Moreover, f 2 A is a zero function on Spec A if and only if f .x/ D 0 for all x 2 Spec A; that is, if and only if f 2 x for all x 2 Spec A. These zero functions have an elegant characterization. Proposition 1.3.2. The following statements are equivalent: (i) The element f 2 A is a zero function on Spec A. (ii) The element f belongs to every prime ideal of A. (iii) The element f is nilpotent. Proof. .ii/ , .iii/. Suppose f is nilpotent. Then f n D 0 for some integer n > 0, so f n 2 x for each x 2 Spec A. Thus either f 2 x or f n1 2 x. Repeating this process (if necessary) shows that f 2 x. For the converse, we suppose f is not nilpotent and show that there exists a prime ideal x for which f 62 x; that is, f 62 \x2Spec A x. Consider the collection I of ideals of A that contain no power of f . This collection is partially ordered by inclusion  and is non-empty since it contains !. Each chain fI˛ g˛2J has an upper bound [˛2J I˛ . Thus, by Zorn’s Lemma, I has a maximal element M . We claim that M is a prime ideal of A. To this end, we show that A=M is an integral domain. Suppose b1 C M and b2 C M are two elements of A=M with b1 ; b2 62 M . We have M  .b1 / C M , so if .b1 / C M contains no power of f , then M D M C .b1 /, which says that b1 2 M , a contradiction. Thus there exists an integer n1 > 0 with f n1 2 .b1 / C M . By similar reasoning, there exists an integer n2 > 0 with f n2 2 .b2 / C M . Now f n1 Cn2 2 .b1 b2 / C M , so if b1 b2 2 M , then f n1 Cn2 2 M , a contradiction. It follows that b1 b2 62 M , and hence b1 b2 6D 0 in A=M . Thus M 2 Spec A with f 62 M since M contains no power of f . It is immediate that .i / , .ii/. t u The presence of zero functions other than 0 2 A is not desirable, and we would like to remove them from consideration. This is done by a simple process. Proposition 1.3.3. Let A be a ring. Then the quotient ring A=nil.A/ is a set of functions on Spec .A=nil.A// in which there is a unique zero function, namely 0 2 A=nil.A/. Proof. In view of Proposition 1.3.2, this amounts to showing that the only nilpotent element in A=nil.A/ is nil.A/. Let a C nil.A/ 2 A=nil.A/, and suppose that .a C nil.A//n D nil.A/ for some n > 0. Then an 2 nil.A/, and thus there exists an integer m > 0 for which .an /m D amn D 0 2 A. Thus a 2 nil.A/, which says that a C nil.A/ D nil.A/. t u As a consequence of Proposition 1.3.3, we can replace A with the ring A=nil.A/, which has no non-trivial nilpotent elements. In practice, we shall assume that A satisfies nil.A/ D 0. Suppose this is the case. Let S  Spec A and let f 2 A. Then the restriction of f to S denoted by fS is a function on S . Moreover, given

1.4 Chapter Exercises

11

f; g 2 A, with gS .x/ 6D 0 for all x 2 S , the function q D fS =gS is defined, with its value at x 2 S given as q.x/ D fS .x/=gS .x/: The function q is a rational function on S . Note that q may not be defined on all of Spec A.

1.4 Chapter Exercises Exercises for 1.1 1. Compute Spec .Z=.60//. 2. Compute Spec .Z Z/. 3. (from the text) Let A be a ring, and let x 2 Spec A. Show that Spec Ax D fyAx W y 2 Spec A; y  xg: p 4. Compute Spec Q. 2/. 5. Suppose  W A ! B is a ring homomorphism with B an integral domain. Show that ker./ 2 Spec A. 6. Prove that ! 2 Spec A if and only if A is an integral domain. 7. Compute Spec QŒt, with t indeterminate. 8. Compute Spec ZV , where V denotes the Klein 4-group. Exercises for 1.2 9. Construct the spectral diagram associated to the inclusion Z ! Z.3/ . 10. Construct the spectral diagram associated to the canonical surjection | W Z ! Z=.10/. 11. Construct the spectral diagram associated to the inclusion R ! RŒt. 12. Construct the spectral diagram associated to the evaluation homomorphism p2 W QŒt ! R. 13. Let  W R ! R0 be a surjection of rings. Prove that a  is an injection. 14. Assume that  W R ! R0 is an injection of rings. Is a  necessarily an injection? 15. Consider the group ring ZC3 with hi D C3 . (a) Show that the ideal .19; 3  2/ is principal. (b) Prove that .p/ factors in ZC3 if 4p1 is the square of an integer. 3 (c) Does the converse of (b) hold? 16. Verify that the spectral diagram in Figure 1.3 is correct. 17. Let R D Z=.20/ and let S D f1; 2; 4; 8; 12; 16g. Construct the spectral diagram associated to the homomorphism Z=.20/ ! S 1 .Z=.20// defined by n 7! n=1.

12

1 The Spectrum of a Ring

18. Let W Spec ZC5 ! Spec Z be the map associated to the inclusion Z ! ZC5 . Compute 1 ..11//. 19. Let  W ZC4 ! ZC2 denote the ring homomorphism defined as g 7! g 2 , hgi D C4 . Construct the map of spectra Spec ZC2 ! Spec ZC4 . Exercises for 1.3 20. Compute the ideal of nilpotent elements in Z=.100/. 21. Show that the set of nilpotent elements in the non-commutative ring nil.Mat2 .Z// does not form an ideal. 22. Give an example of a ring A that has zero divisors and satisfies nil.A/ D f0g. 23. Let S D f.3/; .5/; .7/; : : : ; !g  Spec Z. Compute the collection of all rational functions on S . 24. Let S D f!g  Spec Z. Compute the collection of all rational functions on S . 25. Let S D f.2791/g  Spec Z. Compute the collection of all rational functions on S .

Chapter 2

The Zariski Topology on the Spectrum

The goal of this chapter is to introduce the Zariski topology on Spec A. Throughout this chapter, by “ring” we mean a non-zero commutative ring with unity.

2.1 Some Topology Definition 2.1.1. Let X be an arbitrary set. A topology on X is a collection T of subsets of X for which: (i) ; and X are in T . (ii) The union of the elements of a subcollection of T is in T . (T is closed under arbitrary unions.) (iii) The intersection of the elements of a finite subcollection of T is in T . (T is closed under finite intersections.) The set X together with a topology T is a topological space. A subset U of X is open if U 2 T . A subset V of X is closed if X nV is open. For example, ; is open and X is closed. At the same time, X is open and ; is closed. Example 2.1.1. Let X D R, endowed with the usual absolute value j j, and define T to be the collection of all arbitrary unions of intervals fx 2 R W jx  aj < "g; where a 2 R and " 2 R>0 . Then T is the standard topology on R induced by j j. In the standard topology on R, .2; 4/ D fx 2 R W jx  3j < 1g is open and .1; 2 [ Œ3; 1/ D fx 2 R W jx  3j  1g is closed.

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 2, © Springer Science+Business Media, LLC 2011

13

14

2 The Zariski Topology on the Spectrum

Example 2.1.2. Let X D Q, and let p be a prime number. Let j jp denote the p-adic absolute value defined on Q. Then the j jp -topology on Q is the collection of all arbitrary unions of intervals fx 2 Q W jx  ajp < "g; where a 2 Q and " 2 R>0 . The sequence fan g in Q is a j jp -Cauchy sequence in Q if and only if for each  > 0 there exists a positive integer N for which an is in the open set jan am jp <  whenever m; n > N . We are interested in defining a topology on our set of prime ideals Spec A. Let E be a subset of A, and put V .E/ D fx 2 Spec A W E  xg: Let T be the collection of all subsets U of Spec A such that U D Spec AnV .E/ for some E  A. To show that T is a topology on Spec A, we need two lemmas. Lemma 2.1.1. Let fE˛ g be an arbitrary collection of subsets of A indexed by the set J . Then \˛2J V .E˛ / D V .[˛2J E˛ /. Proof. Let x 2 \˛2J V .E˛ /. Then x 2 V .E˛ / for all ˛ 2 J , and consequently E˛  x for all ˛, which says that [˛2J E˛  x and hence x 2 V .[˛2J E˛ /. Now suppose that x 2 V .[˛2J E˛ /. Then E˛  [˛2J E˛  x for all ˛, which says that x 2 V .E˛ / for all ˛. The lemma follows. t u Lemma 2.1.2. Let fEj g be a finite collection of subsets of A indexed by the set J D f1; 2; : : : ; kg. For each j , 1  j  k, let .Ej / denote the ideal of A generated by the elements of Ej . Then [kj D1 V ..Ej // D V .\kj D1 .Ej //. t u

Proof. Exercise. Proposition 2.1.1. T is a topology on Spec A.

Proof. We show that conditions (i), (ii), and (iii) of Definition 2.1.1 are satisfied. For (i) we have ; D Spec AnV .f0g/ and Spec A D Spec AnV .A/, so ; and Spec A are in T . For (ii) consider an arbitrary subcollection S  T indexed by the set J . We have S D fSpec AnV .E˛ /g˛2J for some collection of subsets fE˛ g˛2J of A. Now, [ \ Spec AnV .E˛ / D Spec An V .E˛ / by DeMorgan’s laws ˛2J

˛2J

D Spec AnV

[ ˛2J

and so (ii) holds.

! E˛

by Lemma 2.1.1,

2.1 Some Topology

15

To prove (iii), consider the finite subcollection S D fSpec AnV .Ei /gkiD1 of subsets of T . We have k \

Spec AnV .Ej / D Spec An

j D1

k [

V .Ej /

j D1

0

D Spec AnV @

k \

by DeMorgan’s laws 1

.Ej /A

by Lemma 2.1.2.

j D1

t u

Thus (iii) holds.

The topology given by Proposition 2.1.1 is the Zariski topology on Spec A. A subset of a topological space can be endowed with a topology in a natural way. Definition 2.1.2. Let X be a topological space with topology T , and let Y be a subset of X . Let TY denote the collection of all subsets of Y of the form fU \ Y W U 2 T g: Then TY is a topology on Y called the subspace topology. A surjective map from a topological space to a set can induce a topology on the set. Definition 2.1.3. Let  W X ! Y be a surjective map, where X is a topological space and Y is an arbitrary set. Let T denote the collection of subsets of Y defined as fU  Y W  1 .U / is open in X g: Then T is a topology on Y called the quotient topology. A topology given a collection of topological spaces can be defined as follows. Definition 2.1.4. Let fX Q˛ g˛2J be a family of topological spaces, and consider the Cartesian product P D ˛2J X˛ . Let Y

U˛

(2.1)

˛2J

be a subset of P , where U˛ is open in X˛ for ˛ 2 J and U˛ D X˛ for all but a finite number of ˛. Let T be the collection of all arbitrary unions of subsets of the form (2.1). Then T is a topology on P called the product topology on P. One can generalize the product topology as follows. Let fX˛ g˛2J be a family of topological spaces, and for ˛ 2 J let Y˛ be an open subset of X˛ Let Y denote the family fY˛ g˛2J .

16

2 The Zariski Topology on the Spectrum

Q Let XY be the subset of ˛2J X˛ consisting of all .x˛ /˛2J for which x˛ 2 Y˛ for all but a finite number of ˛. On XY define a topology by defining the open sets to be the collection of all unions of subsets of the form Y U˛ ; ˛2J

finite number of ˛. We where U˛ is open in X˛ , for all ˛, and U˛ D Y˛ for all but a Q call this topology the restricted product topology on X D ˛2J X˛ with respect to the family Y D fY˛ g. Clearly, the restricted product topology on X with respect to the family fX˛ g˛2J is the ordinary product topology on X . Perhaps the simplest topology we can endow on a set X is the discrete topology. Here, every subset U of X is an open subset of X . In other words, T is the power set of X . Let X and Y be topological spaces. Let W X ! Y be a function from X to Y . Then is continuous if 1 .U / is open in X whenever U is open in Y . For example, if R is endowed with the standard topology, then the map f W R ! R given by x 7! x 2 is continuous. Moreover, if Y is given the quotient topology, then the surjective map of topological spaces  W X ! Y is continuous. Continuous maps can be defined in terms of closed subsets. Proposition 2.1.2. A map W X ! Y is continuous if and only if  1 .V / is closed in X whenever V is closed in Y . Proof. Exercise.

t u

The next proposition shows that the associated map is continuous in the Zariski topology. Proposition 2.1.3. Let  W A ! B be a homomorphism of rings. Endow Spec B and Spec A with the Zariski topology. Then a  W Spec B ! Spec A is continuous. Proof. Put D a , and let V .E/ be a closed set in Spec A. We show that 1 .V .E// is closed in Spec B. Let x 2 1 .V .E//. Then .x/ 2 V .E/, so that E  .x/ D  1 .x/, and thus .E/  x, which says that x 2 V ..E// and therefore 1 .V .E//  V ..E//. Now suppose x 2 V ..E//. Then .E/  x, so that E   1 .x/ D .x/, which yields .x/ 2 V .E/. Thus x 2 1 .V .E//, which gives V ..E//  1 .V .E//. It follows that 1 .V .E// D V ..E//; that is, the pullback of a closed set is closed. t u Under what conditions are two topological spaces essentially the same? We next study the analog of the notion of a group or ring isomorphism. Definition 2.1.5. A map  W X ! Y is a homeomorphism of topological spaces if (i)  is a bijection and (ii)  and  1 are continuous.

2.2 Basis for a Topological Space

17

Two spaces are homeomorphic if there exists a homeomorphism  W X ! Y . We then write X Š Y . For example, if we consider R with the standard topology and endow the subset .1; 1/  R with the subspace topology, then the map f W .1; 1/ ! R defined as x f .x/ D 1x 2 is a homeomorphism of topological spaces. Here is an important example of a homeomorphism. Proposition 2.1.4. Let  W A ! A=nil.A/ be the canonical surjection of rings, and let a  W Spec .A=nil.A// ! Spec A be the associated map of topological spaces. Then a  is a homeomorphism of topological spaces. Proof. We first show that a  is injective. Suppose that x 6D y in Spec .A=nil.A//. Then  1 .x/ 6D  1 .y/, and so a .x/ 6D a .y/, which says that a  is an injection. Now let y 2 Spec A. Then x D y C nil.A/ is a prime ideal of Spec .A=nil.A// with  1 .x/ D y. Thus a .x/ D y, and so a  is a bijection. We already know that a  is continuous (Proposition 2.1.3), so it remains to show that .a /1 W Spec A ! Spec .A=nil.A// is continuous. But this is equivalent to showing that a .V .E// is closed in Spec A for any subset E  A=nil.A/, and this holds since a .V .E// D V .F /, where F is the subset of A for which E D F C nil.A/. t u

2.2 Basis for a Topological Space Another way to describe a topology on a set is to give a set of “building blocks” for the open sets of the topology. Definition 2.2.1. Let X be a topological space. A basis for X is a collection B of open subsets of X that satisfies: (i) For each x 2 X , there exists at least one subset B 2 B for which x 2 B. (ii) For each x 2 B1 \ B2 with B1 ; B2 2 B, there exists a subset B3 2 B for which x 2 B3 and B3  B1 \ B2 . If B is a basis for the topological space X , then the collection of all unions of elements in B is a topology on X called the topology on X generated by B. Of course, the topology on X generated by B coincides with the given topology on X . We are interested in obtaining a basis for the Zariski topology on Spec A. Let f be a non-nilpotent element of A. Then V .ff g/ D fx 2 Spec A W f 2 xg is a closed set in Spec A (since Spec AnV .ff g/ is open by definition). In what follows, we shall write V .f / in place of V .ff g/, and we will set D.f / D Spec AnV .f /. For an ideal I of A, we put D.I / D Spec AnV .I /.

18

2 The Zariski Topology on the Spectrum

The open set D.f / has some important properties. Lemma 2.2.1. Let f; g be non-nilpotent elements of A. Then (i) D.f / \ D.g/ D D.fg/; (ii) D.f / [ D.g/ D D..f; g//. Proof. We prove (i) and leave (ii) as an exercise. Let x 2 D.f / \ D.g/. Then x 2 Spec AnV .f / \ Spec AnV .g/ D Spec An.V .f / [ V .g// by DeMorgan’s laws. Hence, f 62 x and g 62 x. Suppose that fg 2 x. Then, since x is prime, either f 2 x or g 2 x, which is a contradiction. Thus fg 62 x, and so x 62 V .fg/. Hence, x 2 D.fg/. For the converse, suppose that x 2 D.fg/. Then x 62 V .fg/, and so fg 62 x. Now f 2 6 x, for otherwise gf D fg 2 x since x is an ideal. Moreover, by the same reasoning, g 62 x. It follows that x 2 D.f / \ D.g/. t u Lemma 2.2.2. Let f; g be non-nilpotent elements. Then D.f /  D.g/ if and only if f n D ga for some n > 0 and some a 2 A. Proof. Suppose D.f /  D.g/. Then, by DeMorgan’s laws, V .g/  V .f /, which says that if g is contained in the prime ideal x 2 Spec A, then so is f . Now, by Proposition 1.1.6, Spec .A=.g// D fx W x 2 Spec A; g 2 xg; hence f is contained in every prime ideal of A=.g/. By Proposition 1.3.2, f C .g/ is nilpotent in A=.g/. Thus f n 2 .g/ for some integer n > 0; that is, f n D ga, a 2 A. For the converse, suppose f n 2 .g/, n > 0, and let g 2 x. Then f n 2 x, so that f 2 x. Thus V .g/  V .f /, which gives D.f /  D.g/. t u Proposition 2.2.1. The collection B D fD.f / W f is a non-nilpotent element of Ag is a basis for the Zariski topology on Spec A. Proof. We show that conditions (i) and (ii) of Definition 2.2.1 hold. For (i), let x 2 Spec A. Since x 6D A, Anx is non-empty, and thus there exists an element f 2 Anx that satisfies f 62 x. Thus x 62 V .f /. Now, by Proposition 1.3.2, f is non-nilpotent. It follows that x 2 D.f /, where D.f / 2 B. For condition (ii), we suppose that x 2 D.f / \ D.g/ for f; g non-nilpotent. By Lemma 2.2.1(i), D.f / \ D.g/ D D.fg/. We claim that fg is non-nilpotent for otherwise V .fg/ D Spec A by Proposition 1.3.2, and thus Spec AnV .fg/ D ;, which is a contradiction. Thus D.fg/ is an element of B with x 2 D.fg/ and D.fg/ D D.f / \ D.g/. t u Example 2.2.1. A basis for Spec Z is B D fD.f / W f 2 Z; f 6D 0g. As an example of a basis element, we have

2.2 Basis for a Topological Space

19

D.10/ D Spec ZnV .10/ D f.3/; .7/; .11/; : : : ; !g: Every open set D.f / is topologically equivalent to the spectrum of a localized ring. Proposition 2.2.2. The open set D.f / is homeomorphic to Spec Af , where Af is the localization S 1 A, where S is the multiplicative set f1; f; f 2 ; : : : g. Proof. We equip D.f / with the subspace topology induced by Spec A. Spec Af , which is the collection fxAf W x 2 Spec A; f 62 xg, is given the Zariski topology. To prove the proposition, we define a map ' W Spec Af ! D.f / by the rule '.xAf / D x, noting that x 2 D.f /. We show that ' is bijective and bicontinuous. Suppose '.xAf / D '.yAf /. Then x D y and so xAf D yAf , and thus ' is an injection. Next, let y 2 D.f /. Then yAf 2 Spec Af , and so '.yAf / D y, which shows that ' is a bijection. Now let D.fg/ D D.f / \ D.g/ be a basic open set in D.f /. To show that ' is continuous, it is enough to show that ' 1 .D.fg// is open in Spec Af . We have ' 1 .D.fg// D fxAf jx 2 Spec A; f 62 x; g 62 xg D Spec Af nV .g=1/; and so ' is continuous. It remains to show that ' 1 is continuous. Let g=f n be a non-nilpotent element of Af , so that D.g=f n / is a basis element of Spec Af . We show that '.D.g=f n // is open in D.f /. For any xAf 2 D.g=f n /, one has g=f n 62 xAf , which says that g 62 x and f 62 x. Thus '.D.g=f n // D D.fg/ D D.f / \ D.g/; which is open in D.f /.

t u

Definition 2.2.2. A topological space X is Hausdorff if for each pair of distinct points x1 ; x2 2 X there exist open sets U1 ; U2 such that x1 2 U1 , x2 2 U2 , and U1 \ U2 D ;. For example, X D R is Hausdorff in the standard topology. Proposition 2.2.3. Spec Z is not Hausdorff. Proof. Let .p/ and .q/ be two non-trivial points of Spec Z. Suppose that U and W are open sets of Spec Z with .p/ 2 U and .q/ 2 W . Since fD.f /g, f 6D 0, is a basis for Spec Z, we can assume without loss of generality that U D D.m/ and W D D.n/ for some non-zero integers m; n. We claim that D.m/ \ D.n/ 6D ;. Since only a finite number of primes divide a non-zero integer and there are an infinite number of primes, there exists a prime s such that s 6 j m and s 6 j n. Thus .s/ 62 V .m/ [ V .n/. Now, by DeMorgan’s laws,

20

2 The Zariski Topology on the Spectrum

.s/ 2 Spec Zn.V .m/ [ V .n// D Spec ZnV .m/ \ Spec AnV .n/ D D.m/ \ D.n/; and so points in Spec Z cannot be separated by open sets.

t u

Definition 2.2.3. Let X be a topological space. An open covering of X is a collection A of open subsets of X whose union is X . That is, an open covering is a collection A D fU˛ g˛2J , U˛ open in X , that satisfies X D [˛2J U˛ . Definition 2.2.4. X is compact if every open covering A of X admits a finite subcover; that is, X is compact if for every open covering A D fU˛ g˛2J of X there is a finite subcollection C D fUi gkiD1  A with X D [kiD1 Ui . Definition 2.2.5. A subset Y of a topological space X is compact if it is compact in the subspace topology induced by X . Proposition 2.2.4. Let W be a closed subset of the compact topological space X . Then W is compact in the subspace topology induced by X . Proof. Let A D fU˛ \ W g be an open covering of W with U˛ open in X . Now B D fU˛ g [ fX nW g is an open covering of X . Since X is compact, we can choose a finite subcollection fUi gkiD1 [ fX nW g of B that covers X . Now fUi \ W gkiD1 is a subcollection of A that covers W , and so W is compact in the subspace topology. t u Proposition 2.2.5. Let  W X ! Y be a continuous map of topological spaces with X compact. Then .X /  Y is compact in the subspace topology induced by Y . Proof. Let fU˛ \ .X /g˛2J be an open covering of .X / in the subspace topology. Then f 1 .U˛ /g˛2J is an open covering of X , and since X is compact, one can extract a finite subcover f 1 .Ui /gkiD1 of X . Now fUi \ .X /gkiD1  fU˛ \ .X /g˛2J is a finite subcover of .X /. t u The space Spec A with the Zariski topology enjoys the property of compactness. Proposition 2.2.6. Spec A is compact. Proof. Since B D fD.f /; f non-nilpotentg is a basis for Spec A, we can assume that an open covering of Spec A is of the form A D fD.f˛ /g, where ff˛ g˛2J is a set of non-nilpotent elements of A. We have Spec A D [˛2J D.f˛ / with [ ˛2J

and thus

\ ˛2J

D.f˛ / D

[ ˛2J

V .f˛ / D ;. Now

Spec AnV .f˛ / D Spec An

\ ˛2J

V .f˛ /

2.2 Basis for a Topological Space

\

21

V .f˛ / D

˛2J

\

Spec AnD.f˛ /

˛2J

D Spec An

[

D.f˛ /

˛2J

D Spec AnD.I / D V .I /; where I is the ideal of A generated by the set of elements ff˛ g˛2J . Therefore, V .I / D ;. Since V .I / D ;, there is no point x 2 Spec A with I  x. Hence, by Proposition 1.1.1, I D A. Specifically, 1 2 I , and so there is a finite subset ff1 ; f2 ; : : : ; fk g of ff˛ g and ring elements g1 ; g2 ; : : : ; gk such that g1 f1 C g2 f2 C    C gk fk D 1: It follows that I D .f1 ; f2 ; : : : ; fk / D A, and thus V .I / D Spec AnD.I / D Spec An

k [

D.fi /

i D1

D

k \

.Spec AnD.fi //

i D1

D

k \

V .fi / D ;;

i D1

and so Spec A D Spec An

k \ i D1

V .fi / D

k [

.Spec AnV .fi // D

i D1

k [

D.fi /:

i D1

Therefore, fD.fi /gkiD1 is a finite subcollection of fD.f˛ /g that covers Spec A, and thus Spec A is compact. t u Definition 2.2.6. Let S be a subset of a topological space X . The closure of S , denoted by S c , is the intersection of all of the closed sets V that contain S ; that is, S c D \S V V . For example, if .1; 1/ D fx W jxj < 1g  R, then .1; 1/c D Œ1; 1 D fx W jxj  1g, and if f.p/g  Spec Z, then f.p/gc D f.p/g. Observe that a subset S  X is closed if and only if S D S c . We can determine precisely when singleton subsets fxg of Spec A are closed.

22

2 The Zariski Topology on the Spectrum

Proposition 2.2.7. The subset fxg  Spec A is closed if and only if x is a maximal ideal of A. Proof. Suppose fxg  Spec A is closed. Then x D V .E/ for some subset E  A. Let I be an ideal of A for which x  I  A. By Proposition 1.1.1, there exists a prime ideal y for which x  I  y  A. Now, since fxgc D fxg, x D y D I . Thus x is maximal. We leave the converse as an exercise. t u At the other extreme, we have f!g  Spec Z, whose closure is all of Spec Z. These non-closed singleton subsets are of interest. Definition 2.2.7. A point x 2 Spec A for which fxgc D Spec A is a generic point of Spec A. Proposition 2.2.8. Spec A has a generic point if and only if nil.A/ is a prime ideal. Proof. Suppose x 2 Spec A is generic. Then fxgc D Spec A, so that Spec A D \x 2 V V . Hence Spec A D V for all closed V with x 2 V . Thus Spec A D V .x/, which says that x is the nilradical of A. For the converse, suppose nil.A/ is prime. We show that nil.A/ is a generic point of Spec A. Note fnil.A/gc D \nil.A/V V . Now Spec A D V .fnil.A/g/  V for each V containing nil.A/, and thus fnil.A/gc D Spec A. t u For example, ! is a generic point of Spec Z, and ! is a generic point of Spec Fp . Definition 2.2.8. A space X is reducible if there exist proper closed subsets X1  X and X2  X for which X D X1 [ X2 . A non-empty space X is irreducible if it is not reducible. Proposition 2.2.9. The space Spec A is irreducible if and only if it has a generic point. Proof. Suppose x is a generic point of Spec A, and let Spec A D X1 [ X2 be a decomposition. The point x is in either X1 or X2 , so let’s assume x 2 X1 . Then Spec A D fxgc  X1c D X1 . Thus X1 D Spec A and Spec A is irreducible. For the converse, we show that Spec A being irreducible implies the existence of a generic point. We use the contrapositive: we assume that Spec A has no generic point and show that Spec A is reducible. To this end, we assume that nil.A/ is not prime. Then there exist f; g such that fg 2 nil.A/ with both f 62 nil.A/ and g 62 nil.A/. Thus V .fg/ D V .f / [ V .g/ D Spec A with V .f /  Spec A and V .g/  Spec A, so that Spec A is reducible. t u Proposition 2.2.10. Spec A is irreducible if and only if nil.A/ is prime. Proof. Exercise.

t u

For example, since nil.Z  Z/ is not prime, X D Spec .Z  Z/ is reducible. In fact, X D X1 [ X2 with X1 D V .E1 /, E1 D f.n; 0/ W n 2 Zg, and X2 D V .E2 / with E2 D f.0; n/ W n 2 Zg.

2.3 Sheaves

23

2.3 Sheaves We begin this section with some category theory. Definition 2.3.1. A category = is a construction consisting of the following three components: (i) a collection of objects Ob.=/; (ii) for each A; B 2 Ob.=/, a collection of morphisms =.A; B/; (iii) for each A; B; C 2 Ob.=/, a law of composition =.A; B/  =.B; C / ! =.A; C /, where .f; g/ 7! gf , for f 2 =.A; B/, g 2 =.B; C /. Categories exist throughout mathematics. Perhaps the primary example is the category of sets. Here the objects are sets, the morphisms between objects are the functions on the sets, and the law of composition is ordinary function composition. Another important category is the category whose objects are the Abelian groups and whose morphisms are the homomorphisms between them. We assume that a category = satisfies the following axioms. Axiom 1. The collections =.A; B/ and =.C; D/ are disjoint unless A D C and B D D. Axiom 2. If f 2 =.A; B/, g 2 =.B; C /, h 2 =.C; D/, then h.gf / D .hg/f . Axiom 3. For each A 2 Ob.=/, there exists an element IA 2 =.A; A/ for which f IA D f for f 2 =.A; B/ and IA g D g for g 2 =.C; A/. Let X be a topological space, and let = be a category. Definition 2.3.2. A presheaf F on X with values in = is defined by the following data: for each open set U of X , there exists an object F .U / 2 Ob.=/, and for each inclusion U  V of open sets, there exists a morphism %VU W F .V / ! F .U / in =.F .V /; F .U // For which (i) %UU is the identity map and V W (ii) %W U D %U %V whenever U  V  W . We give the following example. Let X be a topological space, and consider X a set of points, forgetting for the moment its topological structure. Let Y be a nonempty set. Let = be the category whose objects are given as Ob.=/ D fMap.W; Y / W W  X g, where Map.W; Y / denotes the set of functions f W W ! Y . For a nonempty open subset U of X , define F .U / to be the collection of functions f W U ! Y , and for U D ; set F .U / to be the unique empty function f W ; ! Y . Moreover, whenever U  V , define a morphism %VU W F .V / ! F .U / by %VU .f / D fU , where fU denotes the restriction of f to U . Then F , together with the restrictions f%VU g, is a presheaf on X that we call the presheaf of ordinary functions on X .

24

2 The Zariski Topology on the Spectrum

For the case X D Spec A, the functions in the presheaf of ordinary functions on X , however, are different from the functions we have already defined on Spec A in 1.3. For example, if F is the presheaf of ordinary functions on X D Spec Z, then F .Spec Z/ is certainly not in a 1-1 correspondence with Z, yet Z is identified with the functions on Spec Z as defined in 1.3. Can we define a presheaf of functions on Spec A that recovers the functions of  1.3? We begin by setting F .;/ D f0g  A. We next define F .U / for a non-empty open set U . Let ff˛ g˛2J denote the collection of non-nilpotent elements of A for which D.f˛ /  U . If there exist indices ˛; ˇ 2 J with D.f˛ /  D.fˇ /, then by Lemma 2.2.2 there exists an integer n˛ > 0 and an element aˇ 2 A for which f˛n˛ D fˇ aˇ ; and thus

aˇ 1 D n˛ : fˇ f˛

Thus, when D.f˛ /  D.fˇ /, there exists a homomorphism Afˇ ! Af˛ defined by raˇl r 7! n l ; fˇl f˛ ˛ which we denote as D.f /

%D.fˇ˛ / W Afˇ ! Af˛ : Q We define F .U / to be the subset of ˛2J Af˛ consisting of all families fu˛ g˛2J for which D.f /

%D.fˇ˛ / .uˇ / D u˛ whenever D.f˛ /  D.fˇ /. In the case where U is the basic open set D.f /, we have F .D.f // D Af . To see this, let fu˛ g be a family in F .D.f //. Then there exists an element u0 2 Af for which D.f /

%D.f˛ / .u0 / D u˛ for all ˛. Thus F .D.f // is identified with Af . By Proposition 2.2.2, Spec Af is homeomorphic to D.f /, and so we write F .Spec Af / D Af . Each element of Af can be written as the rational expression r fn for r 2 A and integer n  0. Moreover, f n 62 x for each x 2 D.f /. Thus r=f n can be viewed as a rational function on D.f / of the type described in 1.3; that is, for each x 2 D.f /, we define

2.3 Sheaves

25

r r.x/ .x/ D n ; fn f .x/ where r.x/ D r .mod x/ and f n .x/ D f n .mod x/. Thus F .Spec Af / D Af is the collection of rational functions on the open set Spec Af . We next show that F .Spec A/ D A. By Proposition 2.2.6, Spec A is compact, and thus there is an open covering fD.fi /gkiD1 of Spec A with A D .f1 ; f2 ;    ; fk /. We assume that no element fi is a unit. Now, for each i , 1  i  k, there exists a prime ideal xi 2 Spec A for which fi 2 xi . (This is a consequence of Proposition 1.1.1.) Thus there are no rational functions on Spec A of the form a=b unless a 2 A and b D 1, and so F .Spec A/ is identified with A. We now see that F defined as above recovers A as the collection of functions on Spec A in the manner described in 1.3. (The statement F .;/ D f0g can now be interpreted as: 0 is the unique function whose domain is ;.) Now that we have an agreeable definition for F .U /, U 6D ;, we suppose U  V and let fu˛ g 2 F .V /. Then D.f˛ /  V . Let fuˇ g denote those components of fu˛ g for which each index ˇ is such that D.fˇ /  U . Since U  V , the family fuˇ g D.f / satisfies %D.fˇ / .u / D uˇ whenever D.fˇ /  D.f /  U . Thus fuˇ g 2 F .U /. In this manner, we define a homomorphism %VU W F .V / ! F .U / by %VU .fu˛ g/ D fuˇ g. Proposition 2.3.1. F , together with the homomorphisms %VU defined above, is a presheaf on Spec A. Proof. We show that F satisfies conditions (i) and (ii) in Definition 2.3.1. By construction, %UU .fu˛ g/ D fu˛ g for all fu˛ g 2 F .U /, so (i) is satisfied. For (ii), suppose there exist open sets U  V  W . Let fuˇ g be the components of fu˛ g 2 F .W / with index ˇ for which D.fˇ /  V ; let fu g be the components of fuˇ g 2 F .V / with index  for which D.f /  U . Then %VU .%W V .fu˛ g// D %VU .fuˇ g/ D fu g D %W .fu g/, so (ii) holds. t u ˛ U The presheaf given in Proposition 2.3.1 is the structure presheaf on Spec A and is denoted by O. A presheaf on X should exhibit some natural properties, which we formalize in the following definition. Definition 2.3.3. The presheaf F on X is a sheaf if, for each open set U  X and each open covering fU˛ g˛2J of U , the following are satisfied: (i) For r; s 2 F .U /, if %UU˛ .r/ D %UU˛ .s/ for all U˛ in the covering, then r D s. (ii) Suppose fr˛ g, r˛ 2 F .U˛ /, is a family of elements indexed over the open sets in U the covering of U . Suppose %UU˛˛ \Uˇ .r˛ / D %Uˇ˛ \Uˇ .rˇ / for all ˛; ˇ. Then there exists an element r 2 F .U / for which r˛ D %UU˛ .r/ for all U˛ .

26

2 The Zariski Topology on the Spectrum

For example, the presheaf of ordinary functions on X is a sheaf. Indeed, suppose that fU˛ g˛2J is a covering of the open set U  X . Suppose that f; g 2 F .U / are such that fU˛ .x/ D gU˛ .x/ for all x 2 U˛ and all U˛ . Then one has f .x/ D g.x/ for all x 2 U . This shows that condition (i) of Definition 2.3.2 holds. Moreover, suppose ff˛ g, f˛ 2 F .U˛ /, is a family of functions indexed over all U˛ in the covering. Suppose .f˛ /U˛ \Uˇ .x/ D .fˇ /U˛ \Uˇ .x/ for all x 2 U˛ \ Uˇ and all U˛ . Then there exists a function f W U ! Y for which f˛ D fU˛ for all U˛ . Thus the presheaf of ordinary functions is a sheaf. Happily, the structure presheaf on Spec A is also a sheaf. Proposition 2.3.2. The structure presheaf O on Spec A is a sheaf of functions. Proof. We restrict our proof to verifying that conditions (i) and (ii) of Definition 2.3.2 hold in the case where U D Spec A. The complete proof can be found in [Sh74, Chapter V, 3]. By compactness, we can assume that an open covering of Spec A is of the form fD.fi /gkiD1 for non-nilpotent elements ffi gkiD1 . Suppose r; s 2 O.Spec A/ D A with Spec A

Spec A

%D.fi / .r/ D %D.fi / .s/ D.1/

D.1/

for all i . Since D.1/ D Spec A, this is equivalent to %D.fi / .r/ D %D.fi / .s/ for all i . But then r=1 D s=1, and so condition (i) holds. For condition (ii), Let fqi g, qi D vi =fin 2 O.D.fi //, be a family of rational D.f / D.f / functions with %D.fii /\D.fj / .qi / D %D.fji /\D.fj / .qj / for all i; j . Since D.fi / \ D.fj / D D.fi fj /, vi fjn .fi fj /n

D

vj fin ; .fi fj /n

and so vi fjn fin fjn D fin fjn vj fin :

(2.2)

Since D.fj / D D.fj2n /, for 1  j  k, the finite covering of Spec A can be written fD.fj2n /gkj D1 , and thus there exist elements g1 ; g2 ; : : : ; gk of A for which Pn Pk 2n n j D1 fj gj D 1. Set q D j D1 vj fj gj . Then, for all i , 1  i  k, fi q D 2n

k X

fi2n vj fjn gj

j D1

D

k X j D1

vi fjn fin fjn gj

by (2.2)

2.4 Representable Functors

27

D vi fin

k X

fj2n gj

j D1

D vi fi ; n

D.1/

and so %D.fi / .q/ D qi for all i , and hence condition (ii) holds.

t u

2.4 Representable Functors In the last section, we introduced the notion of a category. We now ask: What kind of mappings do we have between two categories? Definition 2.4.1. Let = and =0 be categories. A (covariant) functor F W = ! =0 is a rule that assigns to each object X 2 Ob.=/ exactly one object F .X / 2 Ob.=0 / and to each morphism f 2 =.X; Y / exactly one morphism F .f / 2 =0 .F .X /; F .Y // and that satisfies (i) F .gf / D F .g/F .f / for all g 2 =.Y; Z/ and (ii) F .IX / D IF .X / . We can also define mappings between two functors. Definition 2.4.2. Let F and G be functors from the category = to the category =0 . Then a natural transformation W F ! G is a rule that assigns to each object X 2 Ob.=/ a morphism X in =0 .F .X /; G.X // such that for a given morphism f 2 =.X; Y / the diagram X

F .X / ! G.X / F .f /

#

#

G.f /

F .Y / ! G.Y / Y

commutes. Here is an example of a functor that will be of great importance to us. Let R be a ring and let A be a commutative R-algebra with unity, 1A . Let  W R ! A be the R-algebra structure map of A. We assume that .1R / D 1A . For a commutative R-algebra B with unity 1B , let HomR-alg .A; B/ denote the collection of R-algebra homomorphisms f W A ! B; we assume that f .1A / D 1B .

28

2 The Zariski Topology on the Spectrum

Let =R-alg denote the category of commutative R-algebras, where the collection of morphisms =R-alg .A; B/ is given as HomR-alg .A; B/, for A; B 2 Ob.=R-alg /. The law of composition is ordinary function composition. Let =sets denote the category of sets. Fix an object A 2 Ob.=R-alg /, and define a map F W =R-alg ! =sets by the rule F .B/ D HomR-alg .A; B/

(2.3)

for all B 2 Ob.=R-alg /. Proposition 2.4.1. F defined as above is a functor from the category of commutative R-algebras to the category of sets. Proof. Let % 2 =R-alg .S; T / for objects S; T 2 Ob.=R-alg /. Then there exists a unique element F .%/ 2 =sets.F .S /; F .T //, defined as .F .%/.h//.s/ D %.h.s// for h 2 F .S /, s 2 A. So it remains to check that conditions (i) and (ii) of Definition 2.4.1 hold. We prove (i) here and leave (ii) as an exercise. Let  2 =R-alg .T; W / for W 2 Ob.=R-alg /. Then .F .%/.h//.s/ D .%/.h.s// D .%.h.s/// D .F ./.%h//.s/ D F ./.F .%/.h//.s/ D ..F ./F .%//.h//.s/; t u

and so (i) holds.

But how does the functor F defined in (2.3) relate to the sheaf structure of D.f / Spec A? The restriction map %D.g/ W O.D.f // ! O.D.g// of Spec A is a D.f /

homomorphism of R-algebras %D.g/ W Af ! Ag , and thus by the functorality of F there exists a map   D.f / F %D.g/ W F .Af / ! F .Ag / defined as

    D.f / D.f / F %D.g/ .h/ .s/ D %D.g/ .h.s//

for h 2 F .Af /, s 2 A. Put     A D.f / %Afg .h/.s/ D F %D.g/ .h/ .s/:

2.4 Representable Functors

29

Then F is a kind of sheaf on Spec A with Af playing the role of the basic open set A D.f / D.f / and the maps %Afg playing the role of the restrictions %D.g/ . To see this for the case U D Spec A, let fD.fj /gm j D1 be a finite open covering of Spec A. To show that property (i) of Definition 2.3.2 holds, let h; k 2 F .A/, and suppose that A %A Af .h/ D %Af .k/ j

j

for all D.fj / in the covering. Then Spec A

Spec A

%D.fj / .h.s// D %D.fj / .k.s// for all s 2 A, 1  j  m. Thus, by the sheaf property of O, h.s/ D k.s/, and so h D k, which is property (i). Next, let fhj g, hj 2 F .Afj /, be a collection of homomorphisms where j ranges over all of the open sets in the covering. Suppose that Af

Af

%Afi f .hi / D %Afjf .hj / i j

i j

for all i; j . (Here, Afi fj plays the role of D.fi fj / D D.fi / \ D.fj / since Spec Afi fj is homeomorphic to D.fi / \ D.fj /.) Thus, D.f /

D.f /

%D.fii /\D.fj / .hi .s// D %D.fji /\D.fj / .hj .s// for all i; j and s 2 A. Now hi .s/ 2 Afi and hj .s/ 2 Afj , so, by the sheaf property of O, there exists an element hs 2 A, dependent on s, for which Spec A

%D.fi / .hs / D hi .s/ for all i . Thus there exists an element q 2 F .A/, defined by s 7! hs , for which %A Afi .q/ D hi . So property (ii) of Definition 2.3.2 holds. So the functor F of (2.3) corresponds to the structure sheaf O on Spec A. We shall henceforth identify F D HomR-alg .A; / with Spec A, and we say that F is a representable functor that is represented by A. We write RŒF  D A. We next construct a new category. An algebra over R is an R-module A for which the scalar multiplication R  A ! A is given by a ring homomorphism  W R ! A; that is, .r; a/ 7! .r/a for all r 2 R, a 2 A. The map  is the R-algebra structure map of A. We set A D . Let =R denote the category whose objects Ob.=R / consist of all R-algebra structure maps A W R ! A. For objects A ; B 2 =R , the elements of =R .A ; B / consist of ring homomorphisms  W A ! B for which A D B . For such  and r 2 R, a 2 A, .ra/ D .A .r/a/ D .A .r//.a/

30

2 The Zariski Topology on the Spectrum

D B .r/.a/ D r.a/: It follows that =R .A ; B / is in a 1-1 correspondence with =R-alg .A; B/. Let A ; B 2 Ob.=R /. Let A ˝ B denote the tensor product (over R) of the R-modules A and B. Then A ˝ B is an R-algebra with structure map A˝B . Let 1 W A ! A ˝ B, 2 W B ! A ˝ B be morphisms (R-algebra maps) defined as a 7! a ˝ 1, b 7! 1 ˝ b, respectively. As shown in [La84, I,7], .A˝B ; 1 ; 2 / is a coproduct in the category =R . This says that for morphisms W A ! S and W B ! S there exists a unique morphism % W A ˝ B ! S for which the diagram 1 A ! A˝B # % " /  S

2

B

commutes. An important consequence of this is the following. Proposition 2.4.2. Let S be an R-algebra. Then Hom.A; S /  Hom.B; S / D Hom.A ˝ B; S /: Proof. Let 2 HomR-alg .A; S /, 2 HomR-alg .B; S /. There exists an element of f ; 2 HomR-alg .A ˝ B; S / defined as X  X f ; a˝b D .a/ .b/: D f jA˝1 W A ! S and Now suppose f 2 HomR-alg .A ˝ B; S /. Then D f j1˝BPW B ! S are morphisms. There is a morphism f ; W A ˝ B ! S P defined by a ˝ b 7! .a/ .b/. Both f and f ; satisfy the diagram above, and so f D f ; . t u Let  W A ! B be a homomorphism of R-algebras. Then, as we have seen,  determines an associated map a

 W Spec B ! Spec A;

with a .D.f // D f 1 .x/ W x 2 D.f /g for a basic open set D.f / in Spec B.

2.4 Representable Functors

31

Let G and F be the functors represented by B and A, respectively. Then  also determines a map from G to F . Proposition 2.4.3. Let  W A ! B be a homomorphism of R-algebras, and let F and G be the functors defined as F D HomR-alg .A; / and G D HomR-alg .B; /. Then the rule a  that assigns to an object S 2 Ob.=R-alg / the map a

S W G.S / ! F .S /

defined as a S .z/.x/ D z..x// for z 2 G.S /, x 2 A, is a natural transformation of functors a  W G ! F . Proof. Clearly, the map a S is an element of =sets .G.S /; F .S //. Let % W S ! T be an R-algebra homomorphism. Then, for R-algebra map z W B ! S , F .%/.a S .z// D F .%/.z/ D %.z/ D .%z/./ D a T .%z/ D a T .G.%/.z//; and so the diagram a

G.S / G.%/

S !

# G.T /

F .S / #

! T

F .%/

F .T /

a

commutes. Thus a  W G ! F is a natural transformation.

t u

And, with a little more work, one can prove the following. Proposition 2.4.4. [Yoneda’s Lemma] Let G D HomR-alg .B; / and F D HomR-alg .A; / be representable functors from the category of commutative R-algebras to the category of sets. Then the collection of R-algebra homomorphisms A ! B is in a 1-1 correspondence with the collection of natural transformations G ! F . Proof. We show that  7! a  is a bijection between =R-alg .A; B/ and the collection of natural transformations G ! F . Let  W A ! B be an R-algebra homomorphism. Then, by Proposition 2.4.3, a  is a natural transformation. We next show that a natural transformation W G ! F can be written in the form D a  for some R-algebra homomorphism  W A ! B. Since B 2 Ob.=R-alg /, there is a function B W G.B/ ! F .B/. Let IB W B ! B denote the identity map.

32

2 The Zariski Topology on the Spectrum

Put  D B .IB / 2 F .B/. Then  W A ! B is a map of R-algebras. Now, for an R-algebra map z W B ! S , one has S W G.S / ! F .S / with S .z/

D

S .zIB /

D

S .G.z/.IB //

D F .z/.

B .IB //

D F .z/./ D z D a S .z/: Consequently, D a . Thus the map  7! a  is surjective. Suppose that a  D a !. Then a B .IB / D IB  D  and a !B .IB / D IB ! D !. Thus  D !, and so the map  7! a  is a bijection. t u

2.5 Chapter Exercises Exercises for 2.1 1. Let S be a subset of a topological space X . Suppose that for each a 2 S there exists an open set U  S for which a 2 U . Prove that X nS is a closed subset of X . 2. List all of the possible topologies on the set X D fa; b; c; d g. 3. Prove Proposition 2.1.2. 4. Consider the Zariski topology on Spec Z. Determine whether the following subsets are closed, open, neither, or both. (a) f.5/; .7/; .11/; : : : ; !g. (b) f.11/g. (c) f.5/; .7/; .11/; : : : g. 5. Let p be a prime of Z. List all of the open sets in the Zariski topology of Spec .Z=.p//. 6. Suppose Q is endowed with the j jp -topology. Show that the map  W Q ! Q defined as .x/ D x 2 is continuous. Hint: First show that the maps f .x/ D x and g.x/ D x are continuous. 7. Prove that Q in the j jp -topology is not homeomorphic to Q endowed with the subspace topology induced by the standard topology on R. 8. Let  W X ! Y be a map of topological spaces where X has the discrete topology. Prove that  is continuous. Exercises for 2.2 9. Prove that Qp is Hausdorff in the j jp -topology. 10. Show that R is Hausdorff in the discrete topology.

2.5 Chapter Exercises

33

11. Prove that Spec ZŒi is irreducible. What features in the spectral diagram indicate irreducibility? 12. Prove that Spec ZC2 is reducible. What features in the spectral diagram indicate reducibility? 13. Prove that Spec ZC3 is reducible. What features in the spectral diagram indicate reducibility? 14. Prove Lemma 2.2.1(ii). 15. Prove that Spec .Z=.8// is Hausdorff. 16. Compute the closure of S D f.2/; .3/; .5/g in Spec Z. 17. Compute the closure of S D f.3/; .2 C i /; .7/g in Spec ZŒi. 18. Compute the closure of f.1 C /g in Spec ZC2 . 19. Prove the converse of Proposition 2.2.7. 20. Let S D f.1 C i/; .3/; .2 C i/; .7/; .11/; .3 C 2i/; : : : ; g  Spec ZŒi be endowed with the subspace topology. Determine whether S is homeomorphic to Spec Z. 21. Let S D f.2; 1 C /; .2  /; .3  /; : : : ; . pC1  p1 /; : : : ; .1 C /g in 2 2 Spec ZC2 be endowed with the subspace topology. Determine whether S is homeomorphic to Spec Z. Exercises for 2.3 22. Let X be a topological space endowed with the discrete topology, and let A be a non-empty set with at least two elements. Let a 2 A. Let F .;/ D fag and, for each non-empty open subset U of X , let F .U / D A. For non-empty open subsets U , V , U  V , let %VU be the identity function on A; otherwise, for V open, put %V; W F .V / ! fag. Show that F is a presheaf. 23. Prove that the presheaf defined in Exercise 22 is not a sheaf. 24. Let R be endowed with the standard topology, and for an open subset U let F .U / be the collection of all continuous functions  W U ! R. For U  V , let %VU W F .V / ! F .U / be the restriction of  2 F .V / to U . Prove that F is a sheaf. 25. Classify all of the sheaves of sets on Spec .Z=.8//. 26. Suppose that A is a PID, and let O denote the structure presheaf on Spec A. Let U D D.f / [ D.g/, where f; g are non-nilpotent elements of A. (a) Compute O.U /. (b) Prove that O.Spec A/  O.U /  O.D.f //. Exercises for 2.4 27. Finish the proof of Proposition 2.4.1. 28. Let =com denote the category of commutative rings with unity, and let =Abel denote the category of Abelian groups. Define F W =com ! =Abel by the rule F .R/ D hR; Ci. Prove that F is a functor. 29. Let =rings denote the category of rings with unity, and let =gps denote the category of groups. Define F W =rings ! =gps by the rule F .R/ D hU.R/; i. Prove that F is a functor.

34

2 The Zariski Topology on the Spectrum

30. Let X be a topological space, and let =X be the category whose objects are the open sets of X . For U; V 2 Ob.=X / with U  V , the set of morphisms =X .U; V / consists of the inclusion map U  V only. Let =sets denote the category of sets. Show that F is a presheaf of sets on X if and only if F is a contravariant functor from =X to =sets . 31. Let =dom denote the category of integral domains, and let W Z ! Q denote the inclusion homomorphism. Let ˛; ˇ 2 =dom .Q; S /, S 2 Ob.=dom /. Prove that ˛ D ˇ implies ˛ D ˇ.

Chapter 3

Representable Group Functors

Throughout this chapter, by “ring” we mean a non-zero commutative ring with unity.

3.1 Introduction to Representable Group Functors Let A be a commutative R-algebra, and let F be the covariant functor defined as F D HomR-alg .A; /. Let S be an object in Ob.=R-alg /. What structure on A do we need to endow F .S / with a binary operation? Let  W A ! A ˝R A be an R-algebra homomorphism. For a 2 A, we shall write the image of a as X .a/ D a.1/ ˝ a.2/ ; .a/

where a.1/ ; a.2/ 2 A. This is the Sweedler notation for .a/. It is important to note that “.1/”and “.2/”are not subscripts in the usual sense: a.1/ records the left components of the tensors in the expansion of .a/, while a.2/ records the right components in .a/. Since  is an R-algebra homomorphism, .ab/ D .a/.b/ and one writes 0 .ab/ D @

X

10 a.1/ ˝ a.2/ A @

X

.a/

1 b.1/ ˝ b.2/ A D

.b/

X

a.1/ b.1/ ˝ a.2/ b.2/ :

.a;b/

Note that .a.i / /, i  1, is written .a.i / / D

X

a.i / .1/ ˝ a.i / .2/ :

.a.i / /

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 3, © Springer Science+Business Media, LLC 2011

35

36

3 Representable Group Functors

By Proposition 2.4.3, the R-algebra homomorphism  determines a natural transformation of representable functors  W HomR-alg .A ˝R A; / ! HomR-alg .A; /;

a

which by Proposition 2.4.2 can be written as  W F  F ! F;

a

where S W F .S /  F .S / ! F .S /;

a

for an R-algebra S . The morphism aS (binary operation on F .S /) is given by the rule X a S .f; g/.x/ D .f; g/.x/ D f .x.1/ /g.x.2/ / .x/

for f; g 2 F .S /, x 2 A. We set f  g D aS .f; g/: Example 3.1.1. Let A D RG with G a finite Abelian group, and let F D HomR-alg .RG; / be the corresponding functor. Then the map  W RG ! RG ˝R RG defined by X X a  7! a . ˝ /  2G

 2G

is an R-algebra homomorphism and determines a binary operation on F .S / given by .f  g/

X  2G

! a 

D

X

a f ./g./:

 2G

Like all binary operations on sets, those on F .S / can be associative or commutative, admit an identity or inverses, and so forth. Since we have defined a binary operation on F .S / using an algebra map, we can also describe the properties of the binary operation by specifying conditions on this algebra map. We first look at conditions for commutativity and associativity. The map t W A ˝R A ! A ˝R A defined as t.a ˝ b/ D b ˝ a is the twist map. Proposition 3.1.1. Let  W A ! A ˝R A be an R-algebra map that satisfies .a/ D t..a//

(3.1)

for all a 2 A. Then the corresponding binary operation on F .S / D HomR-alg .A; S / is commutative.

3.1 Introduction to Representable Group Functors

37

Proof. We show that .f  g/.a/ D .g  f /.a/ for all a 2 A. We have .f  g/.a/ D .f; g/

X

a.1/ ˝ a.2/

.a/

D .f; g/ D

X

X

a.2/ ˝ a.1/

by (3.1)

.a/

f .a.2/ /g.a.1/ /

.a/

D

X

g.a.1/ /f .a.2/ /

.a/

D .g; f /

X

since S is commutative

a.1/ ˝ a.2/

.a/

D .g  f /.a/:

t u

Let I W A ! A denote the identity map. Let f; g; h 2 F .S /. For a ˝ b ˝ c 2 A ˝ A ˝ A, put .f; g; h/.a ˝ b ˝ c/ D f .a/g.b/h.c/. Proposition 3.1.2. Let  W A ! A ˝R A be an R-algebra map that satisfies .I ˝ /.a/ D . ˝ I /.a/

(3.2)

for all a 2 A. Then the corresponding binary operation on F .S / is associative. Proof. We show that .f  .g  h//.a/ D ..f  g/  h/.a/. Now, .f  .g  h//.a/ D

X

f .a.1/ /.g  h/.a.2/ /

.a/

D

X

f .a.1/ /

.a/

D

X

X

    g a.2/ .1/ h a.2/ .2/

.a.2/ /

    f .a.1/ /g a.2/ .1/ h a.2/ .2/

.a;a.2/ /

D .f; g; h/

X

a.1/ ˝ a.2/ .1/ ˝ a.2/ .2/

.a;a.2/ /

D .f; g; h/

X

.a;a.1/ /

D

X .a;a.1/ /

a.1/ .1/ ˝ a.1/ .2/ ˝ a.2/

    f a.1/ .1/ g a.1/ .2/ h.a.2/ /

by (3.2)

38

3 Representable Group Functors

D

XX

    f a.1/ .1/ g a.1/ .2/ h.a.2/ /

.a/ .a.1/ /

D

X

.f  g/.a.1/ /h.a.2/ /

.a/

D ..f  g/  h/.a/:

t u

What condition on  guarantees the existence of a multiplicative identity element? P Let m W AP˝R A ! A denote the multiplication map of A defined as m. a ˝ b/ D ab, and let  W R ! S denote the structure map of the commutative R-algebra S . We have .1R / D 1S . Proposition 3.1.3. Let  W A ! R be an R-algebra homomorphism for which m.I ˝ /.a/ D a D m. ˝ I /.a/

(3.3)

for a 2 A. Then the R-algebra homomorphism  W A ! S satisfies ./  f D f D f  ./ for all f 2 F .S /. Thus  is a left and right identity for the binary operation on F .S /. Proof. ../  f /.a/ D

X

..a.1/ //f .a.2/ /

.a/

0

Df @

X

1 .a.1/ /a.2/ A

.a/

D f .a/

by (3.3):

In a similar manner, one can show that f D f  ./.

t u

What condition on  guarantees the existence of multiplicative inverse elements? Proposition 3.1.4. Let f 2 F .S /. Let  W A ! A be an R-algebra homomorphism for which m.I ˝ /.a/ D .a/1A D . ˝ I /.a/

(3.4)

for a 2 A. Then the R-algebra homomorphism f  W A ! S satisfies .f /  f D  D f  .f /: Thus f  is a left and right inverse for f with respect to the binary operation on F .S /.

3.1 Introduction to Representable Group Functors

39

Proof. We have ..f /  f /.a/ D

X

f ..a.1/ //f .a.2/ /

.a/

0

Df @

X

1 .a.1/ /a.2/ A

.a/

D f ..a/1A /

by (3.4)

D .a/f .1A / D .a/1S D ..a//: Likewise, one has  D f  .f /.

t u

So we have arrived at the following. Proposition 3.1.5. Let F D HomR-alg .A; / be a functor, together with additional R-algebra maps  W A ! A ˝R A;

 W A ! R;

 W A ! A;

that satisfy conditions (3.2), (3.3), and (3.4), respectively. Then, for each S 2 Ob.=R-alg /, the set F .S / is a group under the binary operation . Proof. As one can easily verify, F .S / together with  satisfies the requirements for F .S / to be a group. t u The functor F in Proposition 3.1.5 is a representable group functor, which is also called an affine group scheme or an R-group scheme. The R-algebra A is the representing algebra of F ; we write RŒF  D A. Note that F is a functor from the category of commutative R-algebras to the category of groups, where the morphisms are homomorphisms of groups. The map  is the comultiplication map of A,  is the counit map of A, and  is the coinverse map of A. When necessary to avoid confusion, we shall denote the comultiplication, counit, and coinverse maps of A by A , A , and A , respectively. Here are some important examples of R-group schemes. The ring R itself as an R-algebra represents the R-group scheme F D HomR-alg .R; /. The comultiplication on R is defined by .1/ D 1 ˝ 1, the counit is defined by .1/ D 1, and the coinverse is given by .1/ D 1. For an R-algebra S , F .S / consists of a single element, the R-algebra structure map  W R ! S , and thus F is the trivial R-group scheme denoted by 1, or more simply 1 when the context is clear. For a non-trivial example, let RŒX  denote the algebra of polynomials in the indeterminate X . Then F D HomR-alg .RŒX ; / is an R-group scheme, with comultiplication defined by .X / D X ˝ 1 C 1 ˝ X , counit defined by .X / D 0, and coinverse given by .X / D X .

40

3 Representable Group Functors

Let us examine this group scheme more closely. Let S be an R-algebra. The group F .S / consists of all R-algebra maps  W RŒX  ! S . These homomorphisms are precisely the evaluation homomorphisms and are determined by sending the indeterminate X to some element a in S . We see that F .S / consists of the algebra maps a W RŒX  ! S , where X 7! a, a 2 S . We have a .1RŒX  / D 1S . Now, how does the group product  in F .S / work? Let a , b be elements of F .S /, and let m W S ˝R S ! S denote the multiplication map of S . Then .a  b /.X / D m.a ˝ b /.X / D m.a ˝ b /.X ˝ 1RŒX  C 1RŒX  ˝ X / D m.a ˝ 1S / C m.1S ˝ b/ D a C b; and so a  b D aCb . We identify F .S / with the additive group S; C of the ring S . For this reason, the group functor F is called the additive R-group scheme, denoted by Ga . For another example, let RŒX1 ; X2  be the R-algebra of polynomials in the indeterminates X1 ; X2 . Let I D .X1 X2  1/, and consider the quotient ring RŒX1 ; X2 =I . There is an isomorphism of R-algebras, f W RŒX1 ; X2 =I ! RŒX; X 1 ; X indeterminate; defined by X1 7! T , X2 7! X 1 . The functor F D HomR-alg .RŒX; X 1 ; / is an R-group scheme with comultiplication  defined by .X / D X ˝ X , counit defined as .X / D 1, and coinverse given as .X / D X 1 . Let us see how this group scheme works. Let S be an R-algebra. The group F .S / consists of the R-algebra maps  W RŒX; X 1  ! S . These maps are determined by sending the variable X to some element .X / in S . But in order for  to be a ring homomorphism, we must have .X 1 / D ..X //1 , and so this element must be a unit of S . We see that F .S / consists of all algebra maps u W RŒX; X 1  ! S , where X 7! u, u 2 U.S /. Now, how is the group product in F .S / defined? Let u , v be elements of F .S /. Then .u  v /.X / D m.u ˝ v /.X / D m.u ˝ v /.X ˝ X / D m.u ˝ v/ D uv: Thus, u  v D uv , and we identify F .S / with the multiplicative group of units in S . This is the multiplicative R-group scheme, which is denoted by Gm .

3.2 Homomorphisms of R-Group Schemes

41

Here are two more examples of R-group schemes. Let A D RŒX =.X n  1/. Then F D HomR-alg .A; / is an R-group scheme with .X / D X ˝ X , .X / D 1, and .X / D X n1 D X 1 . An element  2 F .S / is determined by sending X to an element s in S for which s n D 1. For this reason, F is the multiplicative group of the nth roots of unity, denoted by n . Next, let A D RŒX1;1 ; X1;2 ; X2;1 ; X2;2  be the polynomial algebra in the indeterminates X1;1 ; X1;2 ; X2;1 ; X2;2 . Let J be the principal ideal of A generated by X1;1 X2;1  X1;2 X2;2  1, and consider the quotient ring B D A=J . Then F D HomR-alg .B; / is an R-group scheme where F .S / is the (multiplicative) group of 2  2 matrices M with entries in S with det.M / D 1. This is the special linear group (scheme) of order 2, denoted by SL2 . We leave it as an exercise to formulate the comultiplication, counit, and coinverse maps on the representing algebra B.

3.2 Homomorphisms of R-Group Schemes What are the maps between R-group schemes? Definition 3.2.1. Let W F ! G be a natural transformation of R-group schemes. Then is a homomorphism of R-group schemes if S W F .S / ! G.S / is a homomorphism of groups for all S 2 Ob.=R-alg /. By Yoneda’s Lemma, the homomorphism W F ! G corresponds to an Ralgebra homomorphism  W B ! A with RŒF  D A, RŒG D B. If is a homomorphism of R-group schemes, what can we infer about the map ? By Yoneda’s Lemma, the comultiplication map A W A ! A ˝R A corresponds to the associated map ˛ W HomR-alg .A ˝R A; / ! HomR-alg .A; /. There is a map ˛A˝R A W HomR-alg .A ˝R A; A ˝R A/ ! HomR-alg .A; A ˝R A/. Observe that ˛A˝R A is the group product in HomR-alg .A; A ˝R A/. Let I 2 HomR-alg .A; A/ be the identity map, and let I ˝ I 2 HomR-alg .A ˝R A; A ˝R A/ be defined as .I ˝ I /.a ˝ b/ D a ˝ b. Now, for x 2 B, A˝R A .˛A˝A .I

˝ I //.x/ D ˛A˝R A .I ˝ I /..x// D .I ˝ I /.A ..x/// D A ..x//:

Also by Yoneda’s Lemma, the comultiplication map B W B ! B ˝R B corresponds to the associated map ˇ W HomR-alg .B ˝R B; / ! HomR-alg .B; /. There is a map ˇA˝R A W HomR-alg .B ˝R B; A ˝R A/ ! HomR-alg .B; A ˝R A/; ˇA˝R A is the group operation in HomR-alg .B; A ˝R A/.

42

3 Representable Group Functors

Since

is a group homomorphism, A˝R A .˛A˝A .I

˝ I //.x/ D ˇA˝R A . D.

A .I /

A .I /

˝

˝

A .I //.x/

A .I //.B .x//

D .I ˝ I /. ˝ /.B .x// D . ˝ /.B .x//; and so the R-algebra homomorphism  must satisfy the property A ..x// D . ˝ /B .x/

(3.5)

for all x 2 B. Moreover, since the identity element maps to the identity element under a group homomorphism, A ..x//1R D R .A ..x/// D .R A /..x// D

R .R A /.x/

D .R B /.x/ D B .x/1R ; and so  satisfies A ..x// D B .x/

(3.6)

for all x 2 B. Finally, A .IA A /.x/

which since

A

D .IA A /..x// D A ..x//;

is a group homomorphism equals .

A .IA /B /.x/

D IA .B .x// D .B .x//;

and so  satisfies A ..x// D .B .x//:

(3.7)

So we have arrived at the following characterization of a homomorphism of R-group schemes.

3.2 Homomorphisms of R-Group Schemes

43

Definition 3.2.2. Let W F ! G be a natural transformation of R-group schemes with RŒF  D A, RŒG D B. Then is a homomorphism of R-group schemes if the corresponding map  W B ! A satisfies conditions (3.5), (3.6), and (3.7). Actually, it suffices to show that condition (3.5) holds in Definition 3.2.2. Proposition 3.2.1. Let W F ! G be a natural transformation of R-group schemes with RŒF  D A, RŒG D B. Then is a homomorphism of R-group schemes if the corresponding map  W B ! A satisfies the condition . ˝ /B .x/ D A ..x// for all x 2 B. t u

Proof. Exercise.

Let W F ! G be a homomorphism of R-group schemes with RŒF  D A, RŒG D B. Let  W B ! A denote the corresponding R-algebra homomorphism. Our goal is to arrive at a suitable definition of the kernel of . One would expect ker. / to be a group scheme over R. We know that S W F .S / ! G.S / is a group homomorphism for each S 2 Ob.=R-alg /, and so to describe the kernel of we consider ker. S /. Since B is the identity element of G.S /, we have ker.

S/

D ff 2 F .S / W

S .f

/.x/ D .B /.x/; 8x 2 Bg

D ff 2 F .S / W f ..x// D .B .x//; 8x 2 Bg: Now A can be viewed as a B-algebra with scalar multiplication defined as b  a D .b/a for a 2 A, b 2 B. Likewise, R can be viewed as a B-algebra with scalar multiplication defined by br D B .b/r, and S is a B-algebra with bs D .B .b//s for s 2 S . One has the tensor product A ˝B R, which is also a B-algebra, and the representable functor N D HomB-alg .A ˝B R; /; which is defined on the category of commutative B-algebras. Proposition 3.2.2. Let S be an R-algebra and a B-algebra with scalar multiplication defined as b  s D .B .b//s. Then N.S / D ker. S /. Proof. By Proposition 2.4.2, HomB-alg .A ˝B R; S / D HomB-alg .A; S /  HomB-alg .R; S /: Since HomB-alg .R; S / consists only of the map , N.S / is identified with the elements f 2 HomB-alg .A; S /. We have

44

3 Representable Group Functors

f ..x// D f ..x/1A / D f .x  1A / D x  f .1A / D x  1S D .B .x//1S D .B /.x/: Thus N.S / D ker.

t u

S /.

Thus we have the kernel of described as a representable functor on the category of B-algebras. Our next step is to translate to R-algebras. The augmentation ideal of B, denoted by B C , is the kernel of the counit map B W B ! R. We have the short exact sequence of R-modules 0 ! B C ! B ! R ! 0; and so as R-modules R Š B=B C . But B=B C is also a B-algebra through b  c D bc, and so A ˝B R Š A ˝B .B=B C / Š A=.B C /A as R-algebras. Thus there is a representable functor on the category of commutative R-algebras defined as HomR-alg .A=.B C /A; /: We identify HomR-alg .A=.B C /A; S / with N.S /. (Note that S is viewed simultaneously as an R-algebra and a B-algebra.) We claim that N is an R-group scheme. To prove this, we will need some lemmas. Lemma 3.2.1. Let B be an R-algebra, and let J be an ideal of B. Then there is an isomorphism of R-algebras B=J ˝R B=J Š .B ˝R B/=.J ˝R B C B ˝R J /: Proof. First note that there is an R-algebra map ˛ W B ˝R B ! B=J ˝R B=J; defined as ˛.a ˝ b/ D a ˝ b. Now J ˝R B C B ˝R J  ker.˛/, and so there exists an R-algebra map ˛ W .B ˝R B/=.J ˝R B C B ˝R J / ! B=J ˝R B=J with ˛.a ˝ b/ D a ˝ b.

3.2 Homomorphisms of R-Group Schemes

45

Next, let ˇ denote the canonical surjection of R-algebras ˇ W B ˝R B ! .B ˝R B/=.J ˝R B C B ˝R J / defined as ˇ.a ˝ b/ D a ˝ b. Since ˇ.J ˝ 1/ D ˇ.1 ˝ J / D 0, there exists an R-algebra map ˇ W B=J ˝R B=J ! .B ˝R B/=.J ˝R B C B ˝R J / t defined as ˇ.a ˝ b/ D a ˝ b. Clearly, .˛/1 D ˇ, and thus ˇ is an isomorphism. u We apply Lemma 3.2.1 to the ideal B C to show that B .B C /  B ˝R B C C B ˝R B. C

Lemma 3.2.2. Let B C denote the augmentation ideal of B. Then B .B C /  B˝R B C C B C ˝R B. Proof. Let R W R ! R ˝R R denote the comultiplication map of the trivial group scheme HomR-alg .R; /, and let . ˝ / W B ˝R B ! R ˝R R be the R-algebra map defined by a ˝ b ! .a/ ˝ .b/ for a; b 2 B. For all b 2 B, . ˝ /B .b/ D

X

.b.1/ / ˝ .b.2/ /

.b/

D . ˝ 1/

X

b.1/ ˝ .b.2/ /

.b/

D . ˝ 1/

X

b.1/ .b.2/ / ˝ 1

.b/

D . ˝ 1/.b ˝ 1/

by (3.3)

D .b/ ˝ 1 D R ..b//: Thus, for b 2 B C , . ˝ /B .b/ D R ..b// D 0; and so B .B C /  ker. ˝ /: Now, by Lemma 3.2.1 there is an R-algebra isomorphism ˛ W B=B C ˝R B=B C ! .B ˝R B/=.B ˝R B C C B C ˝R B/;

(3.8)

46

3 Representable Group Functors

which yields a surjective homomorphism of R-algebras ˇ W B ˝R B ! .B ˝R B/=.B ˝R B C C B C ˝R B/ since R ˝R R Š B=B C ˝R B=B C . Consequently, ker. ˝ /  ker.ˇ/ D B ˝R B C C B C ˝R B; and so, by (3.8), B .B C /  B ˝R B C C B C ˝R B.

t u

Lemma 3.2.3. Let W F ! G be a homomorphism of R-group schemes with  W B ! A the corresponding map of R-algebras. Then: (i) A ..B C /A/  .B C /A ˝R A C A ˝R .B C /A. (ii) A ..B C /A/ D 0. (iii) A ..B C /A/  .B C /A. Proof. We prove (i) and leave (ii) and (iii) as exercises. We have A ..B C /A/  A ..B C //.A ˝R A/  .. ˝ /.B .B C ///.A ˝R A/  .. ˝ /.B ˝R B C C B C ˝R B//.A ˝R A/

by Lemma 3.2.2

 .B C /A ˝R A C A ˝R .B C /A:

t u

We are now in a position to show that N is an R-group scheme. Proposition 3.2.3. The representable functor N D HomR-alg .A=.B C /A; / is an R-group scheme. Proof. Let C D A=.B C /A, and put J D .B C /A ˝R A C A ˝R .B C /A. We show that there exist R-algebra maps  W C ! C ˝R C ,  W C ! R, and  W C ! C that satisfy conditions (3.2), (3.3), and (3.4), respectively. First, let A W A ! A ˝R A denote the comultiplication of A, and let ˇ W A ˝R A ! .A ˝R A/=J be the canonical surjection of R-algebras. By Lemma 3.2.3(i), A ..B C /A/  J , and so there exists an R-algebra map ˇA W C ! .A ˝R A/=J: By Lemma 3.2.1, there is an isomorphism ˛ W .A ˝R A/=J ! C ˝R C; and so the map defined as  D ˛ˇA is an R-algebra map  W C ! C ˝R C , which satisfies the condition .I ˝ / D . ˝ I / since A satisfies condition (3.2).

3.3 Short Exact Sequences

47

Next, let A W A ! R be the counit map. By Lemma 3.2.3(ii), there exists an R-algebra map  W C ! R that satisfies, for all c 2 C , .I ˝ /.c/ D c D . ˝ I /.c/; since A satisfies (3.3). Finally, by Lemma 3.2.3(iii), there is a map  W C ! C , induced from A , that evidently satisfies the condition .I ˝ /.c/ D .c/1 D . ˝ I /.c/ for all c 2 C . Thus N D HomR-alg .C; / is an R-group scheme.

t u

Now, we can make the following definition. Definition 3.2.3. Let W F ! G be a homomorphism of R-group schemes, and let  W B ! A denote the corresponding homomorphism of R-algebras. Then the kernel of is the R-group scheme defined as N D HomR-alg .A=.B C /A; /: Let W F ! G be a homomorphism of R-group schemes with ker. / D N . We define an exact sequence of R-group schemes to be the sequence 1!N !F !G with RŒN  D A=.B C /A, RŒF  D A, and RŒG D B. For each S 2 Ob.=R-alg /, the sequence 1 ! N.S / ! F .S / ! G.S / is an exact sequence of groups with N.S / D ker.F .S / ! G.S //.

3.3 Short Exact Sequences In the preceding section, we showed that a homomorphism of R-group schemes gives rise to an exact sequence of R-group schemes. This is analogous to the situation for ordinary abstract groups. The analogy breaks down, however, when we consider short exact sequences. For abstract groups, an exact sequence always extends to a short exact sequence, but this is not the case for R-group schemes. The problem is that the map S 7! F .S /=N.S / is not always an R-group scheme; that is, this map does not necessarily take the form of HomR-alg .Q; / for some R-algebra Q. Let M , L be modules over a ring S , let T be a ring, and let % W S ! T be a ring homomorphism. We will consider T an S -module with s  t D %.s/t.

48

3 Representable Group Functors

Definition 3.3.1. A ring homomorphism % W S ! T is flat if, whenever ˛ W M ! L is an injection of S -modules, the map ' W M ˝S T ! L ˝S T defined as '.m ˝ 1/ D ˛.m/ ˝ 1 is also an injection. As an example, we prove the following. Proposition 3.3.1. Let f be a non-nilpotent element of the ring S . Then the localization map S ! Sf , s 7! s=1, is flat. Proof. Let ˛ W M ! L be an injection of S -modules, and let ' W M ˝S Sf ! L ˝S Sf be the map defined as '.m ˝ 1/ D ˛.m/ ˝ 1. Suppose '.m ˝ 1/ D '.n ˝ 1/, so that ˛.m/ ˝ 1 D ˛.n/ ˝ 1. It follows that ˛.m  n/ ˝ 1 D 0, and so there exists an element f i 2 f1; f; f 2 ; : : : g such that f i ˛.m  n/ D 0. Thus ˛.f i .mn// D ˛.f i mf i n/ D 0, and hence f i m D f i n since ˛ is an injection. Consequently, m ˝ f i D n ˝ f i , and so m ˝ 1 D n ˝ 1 since f i is a unit of Sf . It follows that ' is an injection. t u Lemma 3.3.1. Let % W S ! T be flat, and suppose that P  T D 6 T for every maximal ideal P of S . If M is a non-zero S -module, then M ˝S T is non-zero. Proof. Let m 6D 0 be an element of M , and let I be the annihilator ideal of m. Then S m Š S=I . Since S=I Š S m  M , the flatness of % implies the existence of an injection .S=I / ˝S T ! M ˝S T . Note that .S=I / ˝S T Š T =.I  T /. There exists a maximal ideal P with I  P ; hence T =.I  T / 6D 0 since T =.P  T / 6D 0. It follows that M ˝S T 6D 0. t u Lemma 3.3.2. Let % W S ! T be flat, and suppose that P  T 6D T for every maximal ideal P of S . Let ˛ W M ! N be a map of S -modules, and let ˛ 0 W M ˝S T ! N ˝S T be the induced map defined by m ˝ t 7! ˛.m/ ˝ t. Then, if ˛ 0 is an injection, so is ˛. Proof. Suppose that ˛ W M ! N has non-zero kernel L. Then, by Lemma 3.3.1, L˝S T 6D 0. By the flatness of %, L˝S T ! M ˝S T is an injection. Since L˝S T is in the kernel of ˛0 , ˛ 0 is not an injection, which proves the lemma. t u Definition 3.3.2. A flat map S ! T is faithfully flat if the map ' W M ! M ˝S T defined as '.m/ D m ˝ 1 is an injection for all S -modules M . The localization map S ! Sf may not be faithfully flat, though it can be used to build a faithfully flat map. Let ff1 ; f2 ; : : : ; fn g be a finite set of non-nilpotent elements of S , and suppose that the ideal generated by ff1 ; f2 ; : : : ; fn g is S . Then n Y the map % W S ! Sfi defined as s 7! ..s=1/fi / is faithfully flat. (Prove this as an i D1

exercise. Hint: Use Lemma 3.3.1.) Proposition 3.3.2. Let % W S ! T be faithfully flat. Then % is an injection. Proof. Since S ! T is faithfully flat, the map ' W S ! S ˝S T D T is an injection. u t

3.3 Short Exact Sequences

49

Proposition 3.3.3. Let ˛ W S ! Q be a ring homomorphism, and consider Q an S -module with s  q D ˛.s/q. Suppose S ! T is faithfully flat. Then Q ! Q ˝S T , q 7! q ˝ 1, is faithfully flat. Proof. Let M be a Q-module (also an S -module), and let % W M ! M ˝Q .Q˝S T / be the Q-module map defined by m 7! m ˝ .1 ˝ 1/. There is an isomorphism  W M ˝Q .Q ˝S T / ! M ˝S T defined by m ˝ .q ˝ t/ 7! q  m ˝ t. Now, % W M ! M ˝S T is an injection by the faithful flatness of S ! T . Consequently, % is an injection, and so Q ! Q ˝S T is faithfully flat. t u Proposition 3.3.4. Let S ! T be faithfully flat, and let x 2 Spec S . Then the induced map Sx ! T ˝S Sx is faithfully flat. Proof. Let M be an Sx -module, and let ' W M ! M ˝Sx .T ˝S Sx / be the map defined as m 7! m ˝ .1 ˝ 1/. Note that M is also an S -module with M ˝Sx .T ˝S Sx / Š .M ˝S T / ˝S Sx : Since S ! T is faithfully flat, there is an injection M ! M ˝S T given as m 7! m ˝ 1. Consequently, ' is also an injection. u t Faithful flatness is a critical condition in view of the following. Proposition 3.3.5. Let % W S ! T be a flat ring homomorphism. Then % is faithfully flat if and only if the associated map a % W Spec T ! Spec S is surjective. Proof. Assume that % W S ! T is faithfully flat, and let x 2 Spec S . By Proposition 3.3.4, %x W Sx ! T ˝S Sx is faithfully flat, and therefore Sx =xSx injects into .Sx =xSx / ˝Sx .T ˝S Sx / Š .T ˝S Sx /=.T ˝S xSx /: It follows that xSx D Sx \.T ˝S xSx /. Thus, T ˝S xSx is a proper ideal of T ˝S Sx . By Proposition 1.1.1, T ˝S xSx is contained in a prime ideal J 0 of T ˝S Sx . The preimage of J 0 under the structure map T ! T ˝S Sx is the prime ideal J in Spec T . One then has a %.J / D x, and consequently a % is surjective. For the converse, suppose that a % W Spec T ! Spec S is surjective. Let P be a maximal ideal of S . Then there exists a prime ideal Q 2 Spec T with a %.Q/ D P , and so %.P / D Q with P  T D %.P /T D QT D Q 6D T . Let M be an S -module. Define a map ' W M ˝S T ! .M ˝S T / ˝S T by the rule '.m ˝ t/ D m ˝ 1 ˝ t, and define a map ! W .M ˝S T / ˝S T ! M ˝S T by the rule !.m ˝ t ˝ v/ D m ˝ tv. Then !' is the identity on M ˝S T , and so ' is an injection. An application of Lemma 3.3.2 then implies that M ! M ˝S T , m 7! m ˝ 1, is an injection. Therefore, % W S ! T is faithfully flat. t u If Spec S and Spec T are endowed with the Zariski topology, then faithful flatness is equivalent to the notion that Spec T is an open covering of Spec S . We are now in a position to define surjectivity for homomorphisms of group schemes.

50

3 Representable Group Functors

Definition 3.3.3. The homomorphism of R-group schemes W F ! G is an epimorphism if for each g 2 G.S / there is a faithfully flat R-algebra map % W S ! T for which g 0 2 T .F .T //; where g 0 2 G.%/.g/. Suppose W F ! G is an epimorphism. Since T .F .T // Š F .T /=N.T /, we say that G is the quotient sheaf of F by N and write G D F=N . We define a short exact sequence of R-group schemes to be the sequence 1 ! N ! F ! F=N D G ! 1: If the corresponding map of a homomorphism the homomorphism is an epimorphism.

W F ! G is faithfully flat, then

Proposition 3.3.6. Let W F ! G be a homomorphism of R-group schemes with RŒF  D A, RŒG D B. Suppose that the corresponding algebra map  W B ! A is faithfully flat; that is, suppose that Spec A ! Spec B is surjective. Then is an epimorphism of group schemes. Proof. Let g 2 G.S /. We consider A a B-algebra with b  a D .b/a, for a 2 A, b 2 B, and consider S a B-module with b  s D g.b/s, b 2 B, s 2 S . Let S 0 D S ˝B A denote the tensor product over B. By Proposition 3.3.3, the map S ! S 0 defined by s 7! s ˝ 1 is faithfully flat. Let f 2 HomR-alg .A; S 0 / be defined by a 7! 1 ˝ a. Then f ..b// D 1 ˝ .b/ D 1 ˝ .b/1 D 1˝b1 D b1˝1 D g.b/ ˝ 1: Define a map g 0 W B ! S 0 by b 7! g.b/ ˝ 1. Then an epimorphism.

S 0 .f

/.b/ D g0 .b/, and

is u t

Remark 3.3.1. Let W F ! G be an epimorphism of R-group schemes. Let g 2 G.S / be the trivial element of the group G.S /; that is, suppose g D B . Then S is a B-module with b  s D .B /.b/s. For f 2 HomB-alg .A; S /, f ..b// D f ..b/1/ D f .b  1/ D b  f .1/ D b1 D .B .b//:

3.4 An Example

51

In this case, there exists an element h 2 F .S / D HomR-alg .A; S / with S .h/.b/

D f ..b// D .B .b//;

and so S 0 can be taken to be S . Indeed, we can take any h 2 HomR-alg .A=.B C /A; S /  HomR-alg .A; S /: Thus, the collection of all preimages of B 2 G.S / coincides with the kernel of W F ! G given in Definition 3.2.3.

3.4 An Example In this section, we present an important example of a short exact sequence of group schemes. Let K be a field, let Gm denote the multiplicative group scheme represented by KŒX; X 1 , and let G0m denote a copy that is represented by KŒY; Y 1 . Let m denote multiplication in the K-algebra KŒX; X 1 , and let I denote the identity map on KŒX; X 1 . For an integer l  2, define m.l1/ D m.I ˝ m/.I ˝ I ˝ m/    .I ˝ I ˝    ˝ I ˝m/I ƒ‚ … „ l2

   ˝ I ˝/    .I ˝ I ˝ /.I ˝ /: .l1/ D .I ˝ I ˝ ƒ‚ … „ l2

Then there exists a natural transformation of group schemes p W Gm ! G0m ; where pS W Gm .S / ! G0m .S / is defined by pS .f /.Y / D m.p1/ .f ˝ f ˝    ˝ f /.p1/ .X /: „ ƒ‚ … p

Since .X / D X ˝ X , we have pS .f /.Y / D f .X /p D f .X p /. The K-algebra map corresponding to p is  W KŒY; Y 1  ! KŒX; X 1 , defined by .Y / D X p . Put 0 D KŒY;Y 1  ,  0 D KŒY;Y 1  , and  0 D KŒY;Y 1  . Since . ˝ /0 .Y / D .Y / ˝ .Y / D X p ˝ X p D .X p / D ..Y //; . 0 .Y // D 1 D ..Y //;

52

3 Representable Group Functors

and . 0 .Y // D .Y 1 / D X p D ..Y //; p is a homomorphism of group schemes called the pth power map. The augmentation ideal KŒY; Y 1 C is .Y  1/. Thus the kernel of p is the group scheme N represented by the K-algebra KŒX; X 1 =..Y  1//KŒX; X 1 Š KŒX; X 1 =.X p  1/: Thus there is an exact sequence of K-group schemes p

1 ! p ! Gm ! G0m : In fact, we have the following. Proposition 3.4.1. The map p W Gm ! G0m is an epimorphism of group schemes. Proof. Note that KŒY; Y 1  is the localization M 1 KŒY  at the multiplicative set M D f1; Y; Y 2 ; : : : g. Thus, by Proposition 1.1.4, Spec KŒY; Y 1  consists of ! together with the collection fq.Y /KŒY; Y 1  W q.Y / is irreducible over K and Y 62 q.Y /KŒY g: Let .q.Y // 2 Spec KŒY; Y 1 , q.Y / 6D 0. We have ..q.Y /// D .q.X p //, which is contained in some maximal ideal .r.X // of Spec KŒX; X 1. Now  1 ..r.X /// is a prime ideal of KŒY; Y 1  containing .q.Y //, and hence  1 ..r.X /// D .q.Y //. Thus p..r.X /// D .q.Y //. Moreover, p.!/ D !. Consequently, the map of spectra p W Spec KŒX; X 1  ! Spec KŒY; Y 1  (which we also denote by p) is surjective, and so, by Proposition 3.3.6, p is an epimorphism. u t Let S be an R-algebra and let g 2 G0m .S /. Since p W Gm ! G0m is an epimorphism, there is an R-algebra S 0 and a faithfully flat map S ! S 0 (a Zariski covering Spec S 0 ! Spec S ) for which g has a preimage in Gm .S 0 /. We compute the structure of S 0 . The algebra map g W KŒY; Y 1  ! S is determined by sending Y to a unit a 2 S . There exists a faithfully flat map % W S ! S 0 with S 0 D S ˝KŒY;Y 1  KŒX; X 1. In S 0 , .1 ˝ X /p D 1 ˝ X p D 1 ˝ .Y / D g.Y / ˝ 1 D a ˝ 1:

3.5 Chapter Exercises

53

And so, identifying a with a ˝ 1, one has S 0 Š S ŒT =.T p  a/ for T indeterminate. We have the short exact sequence of group schemes p

1 ! p ! Gm ! G0m ! 1

(3.9)

with Gm = p D G0m . Note that there are elements S 2 Ob.=R-alg / for which pS

1 ! p .S / ! Gm .S / ! G0m .S / ! 1 is not a short exact sequence of abstract groups; that is, there are R-algebras S for which Gm .S /= p .S / 6D .Gm = p /.S /. In Chapter 8, we shall employ short exact sequence (3.9) in the case where K is a field containing Qp .

3.5 Chapter Exercises Exercises for 3.1 1. Referring P to Example P3.1.1, prove that the map  W ZG ! ZG ˝Z ZG defined by .  2G a / D  2G a . ˝ / is a Z-algebra homomorphism. 2. Let F be an R-group scheme, and let S be a commutative R-algebra. Show that the left/right identity element for F .S / is unique. 3. Let R be a commutative ring with unity of characteristic 2. Let F be an Rgroup scheme with RŒF  D A, and let S be a commutative R-algebra. Suppose that f 2 F .S / has order 2 in F .S /, and assume that a 2 A satisfies .a/ D a ˝ 1 C 1 ˝ a. Prove that .a/ D 0. 4. Let F be an R-group scheme represented by the R-algebra A. Suppose that for all a 2 A and ; ˛; ˇ 2 F .S /, X .a/

.a.1/ /˛.a.2/ / D

X

.a.1/ /ˇ.a.2/ /:

.a/

Show that ˛ D ˇ. Exercises for 3.2 5. 6. 7. 8. 9.

Prove Proposition 3.2.1. Prove Lemma 3.2.3, parts (ii) and (iii). Compute the augmentation ideal of RŒGa . Compute the augmentation ideal of RŒGm . Let Gm;Z and Ga;Z denote the multiplicative and additive Z-group schemes, respectively. Prove that W Gm;Z ! Ga;Z defined as S .x/ D 0 for all x 2 Gm;Z .S / is the only homomorphism of Gm;Z into Ga;Z .

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3 Representable Group Functors

Exercises for 3.3 10. Suppose A ! B and B ! C are flat maps of commutative rings. Prove that A ! C is flat. 11. Let A and B be commutative rings. Show that B ! A is faithfully flat if and only if M ˝B A D 0 implies that M D 0 for all B-modules M . 12. Let A and B be commutative rings, and suppose that B ! A is faithfully flat. Show that m  A 6D A for every maximal ideal m of B. p 13. Show that the inclusion Q ! Q. 2/ is faithfully flat. Exercises for 3.4 14. Consider the short exact sequence of Q-group schemes p

1 ! p ! Gm ! G0m ! 1: (a) Find a Q-algebra S for which the sequence p

1 ! p .S / ! Gm .S / ! G0m .S / ! 1 is a short exact sequence of abstract groups. (b) Find a Q-algebra S for which the sequence p

1 ! p .S / ! Gm .S / ! G0m .S / ! 1 fails to be short exact.

Chapter 4

Hopf Algebras

In this chapter, we focus on the structure of the representing algebra of the R-group scheme F .

4.1 Introduction to Hopf Algebras Definition 4.1.1. An R-Hopf algebra is a commutative R-algebra H that is the representing algebra of an R-group scheme F . That is, an R-Hopf algebra is a commutative R-algebra H , together with R-algebra homomorphisms  W H ! H ˝R H WH !R  WH !H

comultiplication; counit; coinverse;

which satisfy, for all a 2 H , the conditions .I ˝ /.a/ D . ˝ I /.a/ coassociativity property; m.I ˝ /.a/ D a D m. ˝ I /.a/ counit property; m.I ˝ /.a/ D .a/1H D m. ˝ I /.a/ coinverse property; P P where m W H ˝R H ! H is multiplication in H defined as m. .a ˝ b// D ab. We will usually denote an R-Hopf algebra by H , but when convenient we will use the notation A, B, or C . Proposition 4.1.1. There is a 1-1 correspondence between the collection of R-group schemes and the collection of Hopf algebras over R.

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 4, © Springer Science+Business Media, LLC 2011

55

56

4 Hopf Algebras

Proof. The map F 7! RŒF  is a bijection between the collection of R-group schemes and the collection of R-Hopf algebras, with inverse H 7! F D HomR-alg .H; /. t u Remark 4.1.1. We should note that one can perfectly well define Hopf algebras that are non-commutative R-algebras. In addition, the requirement that the coinverse be an algebra map can be omitted. A simple example, due to M. Sweedler, is the Q-algebra H D Qh1; g; x; gxi modulo the relations g 2 D 1, x 2 D 0, xg D gx. The comultiplication map  W H ! H ˝Q H is given by g 7! g ˝ g, x 7! x ˝ 1 C g ˝ x, the counit map  W H ! Q is defined as g 7! 1, x 7! 0, and the coinverse map  W H ! H is defined by g 7! g, x 7! gx [Mo93, 1.5.6]. Numerous other non-commutative R-Hopf algebras can be constructed (see, for example, [Mo93, Appendix], where quantum groups are considered). For the purposes of this book, however, we assume that our Hopf algebras are commutative. We do this for three basic reasons. The first is that our notion of Hopf algebra arises as the representing algebra of the structure sheaf SpecA, which is only defined for commutative rings A; in fact, without the assumption of commutativity, we lose the correspondence of Proposition 4.1.1. Second, assuming commutativity facilitates the theory and the computations that follow below. Third, in assuming commutativity, we lose none of the richness of our applications of Hopf algebras that follow. We give some examples of Hopf algebras over R. The ring R itself is an R-Hopf algebra corresponding to the trivial R-group scheme 1; RŒX  is an R-Hopf algebra corresponding to the additive group scheme Ga ; and RŒX; X 1  is an R-Hopf algebra that corresponds to the multiplicative group scheme Gm . As we have seen, if W F ! G is a group scheme homomorphism with algebra map  W B ! A, then ker. / D N D HomR-alg .A=.B C/A; / is a group scheme. Thus A=.B C /A is an R-Hopf algebra. The following is another important example. Example 4.1.1. Let G be a finite Abelian group. Then the R-algebra RG is an R-Hopf algebra with comultiplication  W RG ! RG ˝R RG defined as 

X

! a 

D

 2G

X

a . ˝ /;

 2G

counit  W RG ! R, given by 

X  2G

! a 

D

X  2G

a ;

4.1 Introduction to Hopf Algebras

57

and coinverse  W RG ! RG, defined by 

X  2G

! a 

D

X

a  1 :

 2G

An R-Hopf algebra H is cocommutative if, for all h 2 H , .h/ D .t/.h/, where t is the twist map. The Hopf algebras RŒX , RŒX; X 1 , and RG given above are cocommutative. Cocommutativity is an important property in view of the following proposition. Proposition 4.1.2. Let H be a cocommutative R-Hopf algebra, and let F D HomR-alg .H; / denote the corresponding R-group scheme. Then, for all S 2 Ob.=R-alg /, F .S / is an Abelian group. Proof. This is immediate from Proposition 3.1.1.

t u

Definition 4.1.2. Let H be an R-Hopf algebra with counit map  W H ! R. An integral of H is an element y 2 H that satisfies xy D .x/y for all x 2 H .

R We denote the collection of integrals of H by H . R Proposition 4.1.3. The set of integrals H is an ideal of H . R R Proof. We first show that H is an additive subgroup of H . Let x; y 2 H . Then, for z 2 H , we have z.x C y/ D zx C zy D .z/x C .z/y D .z/.x C y/; R R and soR H is closed under addition. Moreover, 0 2 H since z0 RD 0 D .z/0, and x 2 H since z.x/ D .zx/ D ..z/x/ D .z/.x/. Thus H  H . Now, for w 2 H, w.yz/ D .wy/z D ..w/y/z D .w/.yz/; R t u and thus yz 2 H . As an example, weR consider the group ring RG, where G is a finite Abelian group. We seek to compute RG . X R Proposition 4.1.4. RG D R . P

 2G

P Proof. Let S D R  2G , and let a  2G  2 S , a 2 P R. Since fg2G is an R-basis for RG, an element x 2 RG can be written as x D 2G x  for x 2 R. Now

58

4 Hopf Algebras

X 2G

! x 

a

X

! 

D

 2G

X

x a

2G

D

X



X

a

x

2G

D

!

 2G

!

X

X

!

! 

 2G

x 

a

X

!  ;

 2G

2G

R and so S  RG . R P Now suppose x D  2G x  2 RG . For  2 G, x D ./x D x D

X

x  D

 2G

X

x ./;

 2G

where W G ! G is a permutation of the elements of G. Thus, x D a for some R a 2 R and all  2 G, and so RG  S . t u Definition 4.1.3. Let A, B be R-Hopf algebras and let E, F be the corresponding R-group schemes. Then the R-algebra homomorphism  W B ! A is a homomorphism of Hopf algebras if the corresponding natural transformation W E ! F is a homomorphism of group schemes. Thus  W B ! A is a homomorphism of R-Hopf algebras if . ˝ /B .b/ D A ..b// for all b 2 B. The canonical surjection of R-algebras s W A ! A=.B C /A is a Hopf algebra homomorphism. Indeed, the corresponding natural transformation i W N ! E is a homomorphism of group schemes since iS W N.S / ! E.S / is a group homomorphism for all S . Alternatively, one could show (see 3.2) that ˇA .s.a// D .˛/1 .s ˝ s/A .a/; and so ˛ˇA .s.a// D .s ˝ s/A .a/; 8a 2 A: Let A and B be R-Hopf algebras. The tensor product A ˝R B is an R-algebra with multiplication defined as .a ˝ b/.c ˝ d / D ac ˝ bd . It is also an R-Hopf algebra with comultiplication given as A˝B W A ˝R B ! .A ˝R B/ ˝R .A ˝R B/;

4.1 Introduction to Hopf Algebras

59

where A˝B .a ˝ b/ D .I ˝ t ˝ I /.A ˝ B /.a ˝ b/ 1 0 X D .I ˝ t ˝ I / @ .a.1/ ˝ a.2/ / ˝ .b.1/ ˝ b.2/ /A .a;b/

D

X

.a.1/ ˝ b.1/ / ˝ .a.2/ ˝ b.2/ /:

.a;b/

The counit is .A ˝ B / W A ˝R B ! R ˝R R WD R; with .A ˝ B /.a ˝ b/ D A .a/B .b/, and the coinverse is defined as .A ˝ B / W A ˝R B ! A ˝R B; with .A ˝ B /.a ˝ b/ D A .a/ ˝ B .b/. We leave it as exercises to show that these maps satisfy the coassociativity, counit, and coinverse properties. We consider the case A D B. Proposition 4.1.5. Let A be an R-Hopf algebra. Then multiplication m W A ˝R A ! A is a homomorphism of Hopf algebras. Proof. We first show that m is an R-algebra map. Let a; b; c; d 2 A. Then m..a ˝ b/.c ˝ d // D m.ac ˝ bd/ D acbd D abcd D m.a ˝ b/m.c ˝ d /; and so m is an R-algebra homomorphism. Moreover, .m ˝ m/A˝R A .a ˝ b/ D .m ˝ m/ D

X

X

.a.1/ ˝ b.1/ / ˝ .a.2/ ˝ b.2/ /

.a;b/

a.1/ b.1/ ˝ a.2/ b.2/

.a;b/

D A .ab/ D A .m.a ˝ b//; and so m is a Hopf algebra map.

t u

60

4 Hopf Algebras

Let H be an R-Hopf algebra. Though the coinverse map  W H ! H is an R-algebra homomorphism, it is not an R-Hopf algebra homomorphism. Indeed, let ˛ W HomR-alg .H ˝R H; / ! HomR-alg .H; / be the associated map corresponding to comultiplication on H . The group product in HomR-alg .H; H ˝R H / is given by the map ˛H ˝R H W HomR-alg .H ˝R H; H ˝R H / ! HomR-alg .H; H ˝R H /: Let I 2 HomR-alg .H; H / be the identity map, let I ˝ I 2 HomR-alg .H ˝R H; H ˝R H / be defined as .I ˝ I /.a ˝ b/ D a ˝ b, and let t.I ˝ I / be given as a ˝ b 7! b ˝ a. For x 2 A, . ˝ /.x/ D .I ˝ I /. ˝ /.x/ D ˛H ˝R H .I  ˝ I /.x/ D ˛H ˝R H ..I ˝ I /t/..x// D .I ˝ I /.t..x/// D t..x//: Therefore,  satisfies . ˝ /.x/ D t..x//;

(4.1)

and we say that  is a Hopf antihomomorphism. Of course, if H is cocommutative, then  is a homomorphism of Hopf algebras. To define the notion of a short exact sequence of Hopf algebras, we translate to group schemes. Let W E ! F be a homomorphism of R-group schemes with RŒE D A and RŒF  D B, and suppose that the corresponding homomorphism of Hopf algebras i W B ! A is an injection. The kernel N of is an R-group scheme represented by the R-Hopf algebra A= i.B C /A. Since N ! E is a homomorphism of group schemes, the canonical surjection s W A ! A= i.B C /A is a homomorphism of R-Hopf algebras. There is a short exact sequence of R-modules s

0 ! i.B C /A ! A ! A= i.B C /A ! 0; which is close to what we want but of course cannot be the correct notion of a short exact sequence of Hopf algebras since neither 0 nor i.B C /A has the structure of an R-Hopf algebra. The following is a correct definition. Definition 4.1.4. Let A, B, C be R-Hopf algebras. Let E D HomRalg .A; /, F D HomRalg .B; / be the corresponding R-group schemes. Let 1 denote the

4.1 Introduction to Hopf Algebras

61

trivial R-group scheme represented by R. Suppose s W A ! C is a surjection of Hopf algebras and i W B ! A is an injection of Hopf algebras. The sequence 

i

s



R!B !A!C !R is a short exact sequence of R-Hopf algebras if the homomorphism E ! F corresponding to i W B ! A has a kernel represented by C . That is, the sequence above is short exact if there is a Hopf isomorphism C Š A= i.B C /A. For example, let G be a finite Abelian group, and let W be a subgroup of G. Then the canonical surjection of groups G ! G D G=W induces a surjective homomorphism of R-Hopf algebras s W RG ! RG. Moreover, there is a Hopf inclusion i W RW ! RG. The homomorphism of R-group schemes E ! F , RŒE D RG, RŒF  D RW has a kernel represented by RG= i.RW /C RG D RG. Thus there is a short exact sequence of R-Hopf algebras R ! RW ! RG ! RG ! R:

(4.2)

Generally, let M be an R-module, and let M  D HomR .M; R/ denote the collection of linear functionals on M , which we call the linear dual of M . The linear dual M  is an R-module with scalar multiplication defined as .r f /.v/ D r.f .v// for r 2 R, f 2 M  , and v 2 M . Let  W M ! N be a homomorphism of R-modules. Then there exists a homomorphism of linear duals,   W N  ! M  , defined as   .f /.v/ D f ..v//, for all f 2 N  , v 2 M . If M is free and of finite rank, say m, with basis fb1 ; b2 ; : : : ; bm g, then M  is a free rank m R-module with dual basis ff1 ; f2 ; : : : ; fm g, where fi .bj / D ıij . If M and N are free R-modules of finite rank, then linear duality behaves well with respect to tensor products. Proposition 4.1.6. Let M , N be free R-modules of finite rank m. Then M  ˝R N  Š .M ˝R N / : Proof. Let W M  ˝R N  ! .M ˝R N / be defined as f .a/h.b/ for f 2 M  , h 2 N  , a ˝ b 2 M ˝R N . Then

.f ˝ h/.a ˝ b/ D

.rf ˝ h/.a ˝ b/ D .rf /.a/h.b/ D r.f .a/h.b// D r .f ˝ h/.a ˝ b/ for r 2 R, so is R-linear. We claim that is an isomorphism of R-modules. Let fai g be a basis for M with dual basis ffi g, and let fbi g be a basis for N with dual basis fhi g. Then the collection of tensors ffi ˝ hj g, 0  i; j  m, is a basis for M  ˝R N  . For 0  i; j  m, let ij D

.fi ˝ hj / 2 .M ˝R N / :

62

4 Hopf Algebras

Then ij .al ˝ bm / D fi .al /hj .bm / D ıi l ıj m; and so the images fij g form a basis for .M ˝R N / . It follows that isomorphism of R-modules.

is an t u

Proposition 4.1.7. Let H be a cocommutative R-Hopf algebra that is a free Rmodule of rank m, and let H  D HomR .H; R/ be the R-module of linear functionals on H . Then H  is an R-Hopf algebra. Proof. We first show that H  is a commutative ring. Let H W H ! H ˝R H denote the comultiplication map of H . Then H yields an R-module map H W .H ˝R H / ! H  : By Proposition 4.1.6, there is an isomorphism W H  ˝R H  ! .H ˝R H / ; given as .˛ ˝ˇ/.a ˝b/ D ˛.a/ˇ.b/, and we shall henceforth identify H  ˝R H  with .H ˝R H / . We set mH  D H , and define multiplication on H  as .˛ˇ/.a/ D mH  .˛ ˝ ˇ/.a/ D .˛ ˝ ˇ/H .a/ D

X

˛.a.1/ /ˇ.a.2/ /

.a/

for ˛; ˇ 2 H  , a 2 H . The cocommutativity of H implies that ˛ˇ D ˇ˛, and so H  is a commutative ring. Note that .˛H /.a/ D

X .a/

0 ˛.a.1/ /H .a.2/ / D ˛ @

X

1 a.1/ H .a.2/ /A D ˛.a/;

.a/

and so the unity in H  is H . Moreover, the ring homomorphism H  W R WD   R ! H  , with H  D H , H W R ! H  , endows H  with the structure of a commutative R-algebra. (Here we have identified R D HomR .R; R/ with R.) Let mH W H ˝R H ! H denote multiplication in H . Then multiplication in H ˝R H is given as mH ˝R H D .mH ˝ mH /.I ˝ t ˝ I /. Moreover, multiplication in H  ˝R H  is given as mH  ˝R H  D .mH  ˝ mH  /.I ˝ t ˝ I /. Of course, one has H ˝R H D mH  ˝R H  , where H ˝R H is comultiplication on H ˝R H . Set H  D mH . We claim that H  W H  ! H  ˝R H  is an R-algebra map that satisfies the coassociativity property. For ˛ 2 H  , a; b 2 H , H  .˛/.a ˝ b/ D mH .˛/.a ˝ b/ D ˛.mH .a ˝ b// D ˛.ab/:

4.1 Introduction to Hopf Algebras

63

We have H  .˛ˇ/.a ˝ b/ D .˛ˇ/.ab/ D .˛ ˝ ˇ/.H .ab// D .˛ ˝ ˇ/.H .a/H .b// D .˛ ˝ ˇ/.mH ˝H .H .a/ ˝ H .b// D mH ˝H .˛ ˝ ˇ/.H .a/ ˝ H .b// D .H  ˝ t ˝ H  /.˛ ˝ ˇ/.H .a/ ˝ H .b// D .H  .˛/ ˝ H  .ˇ//.H ˝ t ˝ H /.a ˝ b/ D .H  .˛/ ˝ H  .ˇ//.H ˝H .a ˝ b// D H ˝H .H  .˛/ ˝ H  .ˇ//.a ˝ b/ D mH  ˝H  .H  .˛/ ˝ H  .ˇ//.a ˝ b/ D .H  .˛/H  .ˇ//.a ˝ b/; and so H  is an algebra map. Now, for all a; b; c 2 H , .I ˝ H  /H  .˛/.a ˝ b ˝ c/ D .I ˝ mH /mH .˛/.a ˝ b ˝ c/ D mH .˛/..mH ˝ I /.a ˝ b ˝ c// D mH .˛/.ab ˝ c/ D ˛.mH .ab ˝ c// D ˛..ab/c/ D ˛.a.bc// D ˛.mH .a ˝ bc// D mH .˛/..I ˝ mH /.a ˝ b ˝ c// D .mH ˝ I /mH .˛/.a ˝ b ˝ c/ D .H  ˝ I /H  .a ˝ b ˝ c/; and so H  satisfies the coassociativity property. For a counit map for H  , we let H  D  , where  W H  ! R WD R. Thus H  .˛/.r/ D ˛..r// D ˛.r  1/ D r˛.1/. We have H  .˛ˇ/.r/ D r˛ˇ.1/ D r˛.1/ˇ.1/ D ˇ.1/.r˛.1//

64

4 Hopf Algebras

D ˇ.1/.H  .˛/.r// D .H  .˛/.r//ˇ.1/ D H  .˛/H  .ˇ/.r/; and so H  is an algebra map. We leave it as an exercise to show that H  satisfies the counit property. Finally, we define the coinverse map as H  D H , where H W H  ! H  . We have H  .˛/.a/ D ˛..a//. We leave it as an exercise to show that H  is an algebra map that satisfies the coinverse property. Thus H  is an R-Hopf algebra. u t Proposition 4.1.8. Let H be a cocommutative R-Hopf algebra that is a free Rmodule of finite rank. Let F D HomR-alg .H  ; / be the R-group scheme represented by H  . Then F .R/ D fa 2 H W H .a/ D a ˝ a; a 6D 0g: Proof. It is well-known that H can be identified with the double dual H  . For a 2 H , a.f / D f .a/ for f 2 H  . One has F .R/  H  D H . Now suppose a 2 H with H .a/ D a ˝ a. Then, for f; g 2 H  , a.fg/ D a.H .f ˝ g// D H .f ˝ g/.a/ D .f ˝ g/H .a/ D .f ˝ g/.a ˝ a/ D f .a/g.a/ D a.f /a.g/; so that a is an element of F .R/. Conversely, suppose that a 2 F .R/. Then H .a/.f ˝ g/ D .f ˝ g/H .a/ D a.fg/ D f .a/g.a/ D .f ˝ g/.a ˝ a/ D .a ˝ a/.f ˝ g/: Thus the H .a/ D a ˝ a as linear functionals in H  ˝ H  .

t u

One can dualize a short exact sequence of cocommutative R-Hopf algebras to yield a short exact sequence of duals.

4.2 Dedekind Domains

65

Proposition 4.1.9. Let A, B, and C be cocommutative R-Hopf algebras that are free over R of finite rank. Let 

i

s



R!B !A!C !R be a short exact sequence of R-Hopf algebras. Then there exists a short exact sequence of R-Hopf algebras 

s

i



R ! C  ! A ! B  ! R: Proof. One shows that i  W A ! B  is a Hopf surjection, s  W C  ! A is a Hopf injection, and B  Š A =s  ..C  /C /A as Hopf algebras. t u

4.2 Dedekind Domains We shall soon be interested in studying Hopf algebras over Dedekind domains. For the convenience of the reader, we review some important facts about Dedekind domains. A ring R is Noetherian if every ideal in R is a finitely generated module over R. Q Lemma 4.2.1. Let R be a Noetherian ring, and let li D1 R denote the product of a Q finite number of copies of R. Then every ideal I of li D1 R is finitely generated. Q Proof. I is of the form li D1 Ji for ideals Ji of R. Since each Ji is finitely generated, so is I . t u Lemma 4.2.2. Let R be a Noetherian ring, let M be a finitely generated R-module, and let N be a submodule of M . Then N is finitely generated. Q Proof. Let m1 ; m2 ; : : : ; ml be a generating set for M . Let li D1 R denote the product of l copies of R. Then there is a surjective ring homomorphism W

l Y

R!M

i D1 1 defined Ql as .a1 ; a2 ; : : : ; al / D a1 m1 C a2 m2 C    C al ml . Now  .N / is an ideal in i D1 R that is finitely generated by Lemma 4.2.1. Let fs1 ; s2 ; s3 ; : : : ; sk g denote a generating set for  1 .N /. Then

f.s1 /; .s2 /; : : : ; .sk /g is a generating set for N .

t u

66

4 Hopf Algebras

An integral domain R is integrally closed if R consists of the elements of Frac.R/ that are zeros of monic polynomials over R. Definition 4.2.1. A Dedekind domain is an integrally closed, Noetherian integral domain in which every non-zero prime ideal is maximal. We note that every PID is a Dedekind domain (see [Rot02, Example 11.88]). Let R be a Dedekind domain, and let P be a non-zero prime ideal of R. Let RP denote the localization of R at P . Then RP is a local Dedekind domain with maximal ideal nX o pa W p 2 P; a 2 RP : PRP D Proposition 4.2.1. With the notation above, the ideal PRP is principal. Proof. Let .PRP /1 D fx 2 Frac.RP / W xPRP  RP g: Then .PRP /1 PRP is an ideal of RP with PRP  .PRP /1 PRP  RP : Since PRP is maximal, either PRP D .PRP /1 PRP or .PRP /1 PRP D RP . Since RP is a Noetherian integral domain that is integrally closed and local, the discussion after the proof of [Se79, Chapter I, Lemma 1] applies to show that PRP 6D .PRP /1 PRP . Thus .PRPP /1 PRP D RP . Now, there exist elements ai 2 1 .PRP / and i 2 PRP for which kiD1 ai i D 1. Thus ai i D u for some integer i , 1  i  k, and some unit u in RP . Therefore, a D 1 for some a 2 .PRP /1 , 2 PRP . Let b 2 PRP . Then b D .ba/ with ba 2 RP , and hence PRP is generated by . t u Corollary 4.2.1. Let R be a Dedekind domain, and let P be a non-zero prime ideal of R. Then every non-zero element x 2 RP can be written in the form x D u n , where is a generator for PRP , u 2 RP is a unit in RP , and n  0 is an integer. Proof. By Proposition 4.2.1, . / D PRP for some 2 RP . We claim that \n1 . /n D 0. To this end, let x 2 \n1. /n . Then x D n an , an 2 RP , for n  1. Now 0 D n an  nC1 anC1 D n .an  anC1 /; and so an D anC1 for n  1. Consequently, there is an increasing sequence RP a1  RP a2  RP a3     : Since R is Noetherian, this sequence terminates; that is, RP amC1 D RP am for some m. Thus, amC1 2 RP am , and so amC1 D ram for some r 2 RP . Now, 0 D amC1  ram D amC1  r amC1 D amC1 .1  r /: Since 1r is a unit in RP , amC1 D 0. It follows that x D 0, and so \n1 . /n D 0.

4.2 Dedekind Domains

67

Now suppose that x is a non-zero, non-unit element of RP . Then x 2 . /. Necessarily, there exists an integer n for which x 2 . /n , x 62 . /nC1 . Thus, x D n a for some a 2 RP with a 62 . /; that is, a is a unit in RP . t u The element for which . / D PRP is a uniformizing parameter for RP . Corollary 4.2.2. Let R be a Dedekind domain, and let P be a non-zero prime ideal of R. Then the local ring RP is a PID. Proof. Clearly, f0g and RP are principal. Let be a uniformizing parameter for Rp , and let I be a non-trivial proper ideal of RP . Let S denote the collection of integers n  1 for which n 2 I . Let m be the smallest integer in S . Then . m / D I , so that I is principal. t u Let R be an integral domain with field of fractions K. Then K is an R-module. A fractional ideal of R is a non-zero R-submodule J of K of the form J D cI; where c 2 K  and I is an ideal of R. For example, ZŒ 12  D fn=2 W n 2 Zg is a fractional ideal of Z. Proposition 4.2.2. Let R be a Dedekind domain. Then a non-zero submodule J of K is a fractional ideal if and only if it is finitely generated. Proof. Let J be a non-zero submodule of K of the form cI , c 2 K. Then J is finitely generated since I is finitely generated. Conversely, suppose J is a non-zero submodule of K that is finitely generated as an R-module. Write J D Rq1 C Rq2 C    C Rql for q1 ; q2 ; : : : ; ql 2 K. There is a generating set for J of the form fa1 =q; a2 =q; : : : ; al =qg for some ai 2 R, q 2 K. Consequently, J D cI , where c D q 1 and I D .a1 ; a2 ; : : : ; al /. t u The ring of integers in a finite extension of number fields is a Dedekind domain. We spend the remainder of this section proving this fact. We begin with some lemmas. Lemma 4.2.3. Let K be a finite extension of Q, and let R be the ring of integers in K. Then every non-zero ideal J of R contains a basis for K over Q. Proof. Let ˇ 2 K, ˇ 6D 0. Then the set f1; ˇ; ˇ 2 ; : : : ; ˇ l g, l D ŒK W Q, is linearly dependent over Q, and so there exist integers ai 2 Z, 1  i  l, not all zero, for which a0 C a1 ˇ C a2 ˇ 2 C    C al ˇl D 0: Let j be the largest index such that aj 6D 0. Then j 1

aj

.a0 C a1 ˇ C a2 ˇ2 C    C aj ˇj / D 0;

68

4 Hopf Algebras

and so j 1

aj

j 2

a0 C aj

j 3

a1 .aj ˇ/ C aj

a2 .aj ˇ/2 C    C .aj ˇ/j D 0I

thus, aj ˇ is integral over Z. Hence aj ˇ 2 R. Next, let fb1 ; b2 ; : : : ; bl g be a basis for K over Q. Now, by the preceding paragraph, there exist integers ci for which fci bi gli D1  R. Let a be a non-zero element of J . Then fci bi agli D1  J is a basis for K=Q. t u Next, we fix a non-zero ideal J of R and let S be the collection of all bases B for K=Q that are contained in J . By Lemma 4.2.3, S is nonempty. For each basis B D fb1 ; b2 ; : : : ; bl g in S, let NB D Zb1 ˚ Zb2 ˚    ˚ Zbl  J be the free Z-module with basis B D fbi g. For a; b 2 K and 1  i  l, one has abbi D

l X

qi;j bj

j D1

for qi;j 2 Q. The map B W K  K ! Q defined as B.a; b/ D

l X

qi;i

i D1

is a symmetric non-degenerate bilinear form on K. We compute the discriminant of NB with respect to B. We denote this discriminant as disc.NB /. Since B W R  R ! Z, disc.NB / is generated by a non-zero integer that we identify with disc.NB /. The collection fjdisc.NB /jgB2S is a non-empty set of positive integers and as such has a smallest element. Let M D fm1 ; m2 ; : : : ; ml g denote the basis in S that corresponds to the smallest integer jdisc.NM /j. Lemma 4.2.4. Let M D fm1 ; m2 ; : : : ; ml g denote the basis in S that corresponds to the smallest integer jdisc.NM /j. Then J D NM . Proof. We only need to show that J  NM . Let a 2 J . Since fmi g is a basis for K=Q, a D q1 m1 C q2 m2 C    C ql ml for elements qi 2 Q. We claim that each qi is an integer. By way of contradiction, let’s assume that qj 62 Z for some j . Without loss of generality, we can assume that j D 1. Note that q1 D C for some 2 Z and  with 0 <  < 1. Set m01 D a m1 and m0i D mi for 2  i  l. Then M0 D fm0i g is a basis for K=Q that is contained in J .

4.2 Dedekind Domains

69

Let NM0 D Zm01 ˚ Zm02 ˚    ˚ Zm0l . Then NM0  NM . The matrix that multiplies the basis fmi g to give the basis fm0i g is 1  q2 q3    ql B0 1 0 0C C B B0 0 1 0C C DB C: C B: : : : @: :A 0

0 0



0

1

Now, by a familiar property of discriminants, disc.NM0 / D .det.C //2 disc.NM / D .2 Z/disc.NM /; which contradicts the minimality of jdisc.NM /j since 2 < 1. Thus each qi is an integer, and so J D NM . u t Lemma 4.2.5. Let J be a non-zero ideal of R. Then R=J is a finite ring. Proof. Let a 2 J \ Z>0 . (Why is J \ Z>0 non-empty?) Since the ring homomorphism R=.a/ ! R=J is surjective, we only need to show that R=.a/ is finite. By Lemma 4.2.4, J D Zm1 ˚ Zm2 ˚    ˚ Zml for some elements mi 2 J . Now S D fa1 m1 C a2 m2 C    C al ml W 0  ai  ag is a set of coset representatives for R=.a/. Note that jS j D al < 1, so that jR=.a/j D al . t u Lemma 4.2.6. Let F be a non-empty collection of ideals of R. Then every nonempty set of ideals of R has a maximal element. Proof. Let I0 be an ideal in F , and suppose there is an ideal I1 2 F with I0  I1 (proper inclusion). Next suppose there is an ideal I2 with I1  I2 . Continuing in this manner, if there is always an ideal Ii C1 for which Ii  Ii C1 , then I0  I1  I2     is an increasing chain of ideals; thus R=I0 has an infinite number of ideals, which contradicts Lemma 4.2.5. t u Lemma 4.2.7. Let I be an ideal of R. Then I is finitely generated. Proof. Let F denote the collection of all finitely generated ideals that are contained in I . Since f0g 2 F , F is non-empty. By Lemma 4.2.6, F contains a maximal element M . We have M  I . If M  I , then there exists an element a 2 I nM .

70

4 Hopf Algebras

Let J D M C Ra. Then J 2 F and M  J , which contradicts the maximality of M . Hence M D I and I is finitely generated. t u Proposition 4.2.3. Let K be a finite extension of Q with ring of integers R. Then R is a Dedekind domain. Proof. We show that the conditions of Definition 4.2.1 hold. Certainly R is integrally closed by the definition of the ring of integers; moreover, R is Noetherian by Lemma 4.2.7. Let P 6D 0 be a prime ideal of R. By Lemma 4.2.5, R=P is a field. Thus P is maximal. t u

4.3 Hopf Modules The purpose of this section is to establish a fundamental result of R. Larson and M. Sweedler [LS69] that states that for a Dedekind domain R and cocommutative  R-Hopf algebra H that R is free and of finite rank R over R, the linear dual H is isomorphic to H ˝R H  as H -modules, where H  is the ideal of integrals of H . Definition 4.3.1. Let H be an R-Hopf algebra, and let M be an R-module. Then M is a left H-comodule if there exists an R-linear map ‰ W M ! H ˝R M for which (i) .I ˝ ‰/‰ D . ˝ I /‰ and (ii) . ˝ I /‰.m/ D 1 ˝ m; 8m 2 M . One easily checks that H is a left comodule over itself with the comultiplication map  playing the role of ‰. Let M be a left H -comodule with structure map ‰ W M ! H ˝ M . We adapt the Sweedler notation to write X b.1/ ˝ ˇ.2/ ‰.ˇ/ D .ˇ/

for ˇ; ˇ.2/ 2 M , b.1/ 2 H . Observe that X ‰.ˇ.2/ / D b.2/ .1/ ˝ ˇ.2/ .2/ : .ˇ.2/ /

We can extend the Sweedler notation as follows. Let ˇ 2 M . Then 0 1 X b.1/ ˝ ˇ.2/ A .I ˝ ‰/‰.ˇ/ D .I ˝ ‰/ @ .ˇ/

D

X .ˇ;ˇ.2/ /

b.1/ ˝ b.2/ .1/ ˝ ˇ.2/ .2/

(4.3)

4.3 Hopf Modules

71

and

0 . ˝ I /‰.ˇ/ D . ˝ I / @

X

1 b.1/ ˝ ˇ.2/ A

.ˇ/

D

X

b.1/ .1/ ˝ b.2/ .2/ ˝ ˇ.2/ :

(4.4)

.ˇ;b.1/ /

By Definition 4.3.1(i), .I ˝ ‰/‰ D . ˝ I /‰, and so the expressions in (4.3) and (4.4) are equal. The common value in (4.3) and (4.4) will be denoted as X b.1/ ˝ b.2/ ˝ ˇ.3/ : .ˇ/

Similarly, the common value of .I ˝ I ˝ ‰/.I ˝ ‰/‰.ˇ/ D .I ˝ I ˝ ‰/. ˝ I /‰.ˇ/ D .I ˝  ˝ I /. ˝ I /‰.ˇ/ D .I ˝  ˝ I /.I ˝ ‰/‰.ˇ/ D . ˝ I ˝ I /.I ˝ ‰/‰.ˇ/ D . ˝ I ˝ I /. ˝ I /‰.ˇ/ is denoted as

X

b.1/ ˝ b.2/ ˝ b.3/ ˝ ˇ.4/ :

.ˇ/

Definition 4.3.2. Let H be an R-Hopf algebra. The R-module M is a left Hopf module over H if (i) M is a left H -module, (ii) M is a left H -comodule with structure map ‰ W M ! H ˝R M , and (iii) the left H -comodule structure map ‰ W M ! H ˝R M is a left H -module map, where H ˝R M is a left H -module with scalar multiplication X h.1/ k ˝ .h.2/  m/: h  .k ˝ m/ D .h/

Let R be a Dedekind domain, and let H be a cocommutative R-Hopf algebra that is a free R-module of rank n. By Proposition 4.1.7, H  is an R-Hopf algebra that is R-free of rank n. The main results of this section are to prove that H  is a R  left Hopf module over H and that H Š H ˝R H  as H -modules. We start by viewing H  as a left H -module with scalar product defined as X ˇ.1/ ..h//ˇ.2/ .k/ (4.5) .h  ˇ/.k/ D ˇ..h/k/ D .ˇ/

72

4 Hopf Algebras

for h; k 2 H , ˇ 2 H  . In fact, H  is a left H -module algebra (see the following lemma). Lemma 4.3.1. Let h 2 H , ˇ; ˛ 2 H  . Then X h  .ˇ˛/ D .h.1/  ˇ/.h.2/  ˛/: .h/

Proof. Since the comultiplication H  is an R-algebra homomorphism, .h  .ˇ˛//.k/ D .ˇ˛/..h/k/ X D .ˇ.1/ ˛.1/ /..h//.ˇ.2/ ˛.2/ /.k/ .ˇ;˛/

for all k 2 H . Moreover, ..h// D

X

.h.1/ / ˝ .h.2/ /; thus,

.h/

X

.ˇ.1/ ˛.1/ /..h//.ˇ.2/ ˛.2/ /.k/

.ˇ;˛/

D

X

ˇ.1/ ..h.1/ //˛.1/ ..h.2/ //.ˇ.2/ ˛.2/ /.k/

.ˇ;˛;h/

D

X

ˇ.1/ ..h.1/ //˛.1/ ..h.2/ //ˇ.2/ .k.1/ /˛.2/ .k.2/ /

.ˇ;˛;h;k/

D

X

ˇ.1/ ..h.1/ //ˇ.2/ .k.1/ /˛.1/ ..h.2/ //˛.2/ .k.2/ /

.ˇ;˛;h;k/

D

X

ˇ..h.1/ /k.1/ /˛..h.2/ /k.2/ /

.h;k/

D

X

.h.1/  ˇ/.k.1/ /.h.2/  ˛/.k.2/ /

.h;k/

D

X ..h.1/  ˇ/.h.2/  ˛//.k/; .h/

t u

which proves the lemma. 



The left H -module H is also a right H -module with scalar product taken to be multiplication on the right in H  . At the same time, one can define a left H comodule structure on H  as follows. Let f˛i ; biP gniD1 , ˛i 2 H  , bi 2 H , be a   dual basis for H; H . For ˇ 2 H , we have ˇ D niD1 bi .ˇ/˛i . Now H  is a left H -comodule with structure map ‰ W H ! H ˝ H

4.3 Hopf Modules

73

defined as ‰.ˇ/ D

n X

bi ˝ ˇ˛i :

(4.6)

i D1

This H -comodule structure induces a right H  -module structure on H  defined as .ˇ/˛ D m.˛ ˝ I /‰.ˇ/ D

m X

˛.bi /ˇ˛i :

i D1

The surprising result is that the action .ˇ/˛ is precisely the right multiplication action of H  on itself since .ˇ/˛ D

m X

˛.bi /ˇ˛i D ˇ

i D1

m X

bi .˛/˛i D ˇ˛:

i D1

As a consequence of this, one has the useful formula (in Sweedler notation) ˇ˛ D

X

˛.b.1/ /ˇ.2/ :

(4.7)

.ˇ/

So we consider H  as both a left H -module through (4.5) and a left H -comodule through (4.6). In addition, there is a left H -module structure on H ˝ H  defined by the comultiplication on H h.k ˝ ˇ/ D

X

h.1/ k ˝ .h.2/  ˇ/

(4.8)

.h/

for h; k 2 H , ˇ 2 H  . Proposition 4.3.1. (Larson and Sweedler) Let R be a Dedekind domain and let H be a cocommutative R-Hopf algebra that is free and of finite rank n over R. Let H  be an H -module through (4.5), and let H ˝ H  be an H -module through (4.8). Then the map ‰ W H  ! H ˝ H  (as in (4.6)) is a homomorphism of H -modules; that is, H  is a Hopf module over H .  Proof. We show that Pn ‰.h  ˇ/ D h‰.ˇ/ for h n2 H , ˇ 2 H . Recall that, for  ˇ 2 H , ‰.ˇ/ D i D1 bi ˝ ˇ˛i , where f˛i ; bi gi D1 is a dual basis for H ; H . For ˛ 2 H  , one has 00 1 1 X .h.1/ /h.2/ A  ˇ A ˛ counit prop. .h  ˇ/˛ D @@ .h/

74

4 Hopf Algebras

D

X .h.2/  ˇ/..h.1/ /1H  ˛/ .h/

X D .h.3/  ˇ/..h.1/ /h.2/  ˛/ .h/

by the coinverse property. Now, for k 2 H , ..h.1/ /h.2/  ˛/.k/ D ˛.h.1/ .h.2/ /k/

by [Ch00, Proposition 1.11]

D ˛.1/ .h.1/ /˛.2/ ..h.2/ /k/ D ˛.1/ .h.1/ /.h.2/  ˛.2/ /.k/: Thus, .h  ˇ/˛ D

X

˛.1/ .h.1/ /.h.3/  ˇ/.h.2/  ˛.2/ /

.h;˛/

D

X

˛.1/ .h.1/ /.h.2/  .ˇ˛.2/ // by Lemma 4.3.1

.h;˛/

D

X

11 0 n X ˛.1/ .h.1/ / @h.2/  @ ˛.2/ .bj /ˇ˛j AA 0

D

X

˛.1/ .h.1/ /

n XX

n X

˛.2/ .bj /.h.2/  .ˇ˛j //

j D1

.h;˛/

D

by (4.7)

j D1

.h;˛/

˛.1/ .h.1/ /˛.2/ .bj /.h.2/  .ˇ˛j //

.h;˛/ j D1

D

n XX

˛.h.1/ bj /.h.2/  .ˇ˛j //:

.h/ j D1

It follows that bi ˝ .h  ˇ/˛i D bi ˝

n XX

˛i .h.1/ bj /.h.2/  .ˇ˛j //

.h/ j D1

D

n XX .h/ j D1

bi ˛i .h.1/ bj / ˝ h.2/  .ˇ˛j /:

4.3 Hopf Modules

75

Thus ‰.h  ˇ/ D

n X

bi ˝ .h  ˇ/˛i

i D1

D

n XX n X

bi ˛i .h.1/ bj / ˝ .h.2/  .ˇ˛j //

i D1 .h/ j D1

D

n X n XX

bi ˛i .h.1/ bj / ˝ .h.2/  .ˇ˛j //

.h/ i D1 j D1

D

n XX

h.1/ bi ˝ h.2/  .ˇ˛i /

.h/ i D1

Dh

n X

! bi ˝ ˇ˛i

i D1

D h‰.ˇ/:

t u

R  We next show that R H Š H ˝R H  as H -modules. We begin with the following characterization of H  . R Lemma 4.3.2. H  D W; where W D fˇ 2 H  W ‰.ˇ/ D 1 ˝ ˇg. R Proof. Let ˇ 2 H  . Then, for all ˛ 2 H  , ˛ˇ D H  .˛/ˇ D ˛.1/ˇ. Thus, ˛i ˇ D ˇ˛i D ˛i .1/ˇ; where f˛i ; bi g is a dual basis. Thus, ‰.ˇ/ D

n X

bi ˝ ˇ˛i

i D1

D

n X

bi ˝ ˛i .1/ˇ

i D1

D

n X

˛i .1/bi ˝ ˇ

i D1

D 1 ˝ ˇ; and thus ˇ 2 W . Conversely, if ˇ 2 W , then, for all ˛ 2 H  ,

by (4.7), and so ˇ 2

R

H.

˛ˇ D ˛.1/ˇ D H  .˛/ˇ t u

76

4 Hopf Algebras

For ˛ 2 H  , let

X

.˛/ D

.a.1/ /  ˛.2/ :

.˛/

Lemma 4.3.3. ‰. .˛// D 1 ˝ .˛/. Proof. We have 0 ‰. .˛// D ‰ @

X

1 .a.1/ /  ˛.2/ A

.˛/

D

X

.a.1/ /‰.˛.2/ /

by Proposition 4.3.1

.˛/

D

X

.a.1/ /.a.2/ ˝ ˛.3/ /

.˛/

D

X

.a.2/ /a.3/ ˝ ..a.1/ /  ˛.4/ /

 is a Hopf homomorphism

.˛/

D

X .˛/

D 1˝

.a.2/ /1 ˝ ..a.1/ /  ˛.3/ / X

..a.2/ /a.1/ /  ˛.3/

.˛/

D 1˝

X

.a.1/ /  ˛.2/

.˛/

D 1 ˝ .˛/:

t u R

Now, by Lemma 4.3.2, .H  / 

H.

Moreover, we have the following.



Lemma 4.3.4. For h 2 H , .h  H / D .h/ .H  /. Proof. Let ˛ 2 H  . By Proposition 4.3.1, ‰.h  ˛/ D

X

h.1/ a.1/ ˝ .h.2/  ˛.2/ /:

.h;˛/

Thus .h  ˛/ D

X

.h.1/ a.1/ /  .h.2/  ˛.2/ /

.h;˛/

D

X

..a.1/ /.h.1/ /h.2/  ˛.2/ /

.h;˛/

4.3 Hopf Modules

77

D .h/

X

.a.1/ /  ˛.2/

.˛/

D .h/ .˛/

t u

R

Now, H ˝R H  is a left H -module with scalar multiplication defined as h  .k ˝ ˛/ D hk ˝ ˛. This R is precisely the restriction ofRthe H -module structure of (4.8) to the subset H ˝ H  . Indeed, for h; k 2 H , ˛ 2 H  , h  .k ˝ ˛/ D

X

h.1/ k ˝ h.2/  ˛

.h/

D

X

h.1/ k ˝ .h.2/  1H  /˛

.h/

D

X

h.1/ k ˝ H  .h.2/  1H  /˛

.h/

D

X

h.1/ k ˝ .h.2/  1H  /.1/˛

.h/

D

X

h.1/ k ˝ H .H .h.2/ //˛

H D 1H 

.h/

D

X

h.1/ k ˝ H .h.2/ /˛

[Swe69, Proposition 4.0.1]

.h/

D

X

h.1/ H .h.2/ /k ˝ ˛

.h/

D

X

hk ˝ ˛:

.h/

Since .H  / 

R

H,

there exists a map % W H  ! H ˝

%.˛/ D .I ˝ /‰.˛/ D

X

R H

defined as

a.1/ ˝ .˛.2/ /:

.˛/

Proposition 4.3.2. (Larson and Sweedler) Let H  be an H -module through (4.5), R and let H ˝ H  be a left H -module with scalar R multiplication defined as h  .k ˝ ˛/ D hk ˝ ˛. Then the map % W H  ! H ˝ H  is an isomorphism of H -modules. R Proof. We show that % is an H -module homomorphism. Let h 2 H , ˛ 2 H . Then %.h  ˛/ D .I ˝ /‰.h  ˛/ X D .I ˝ / h.1/ a.1/ ˝ .h.2/  ˛.2/ / by Prop. 4.3.1 .h;˛/

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4 Hopf Algebras

D

X

h.1/ a.1/ ˝ .h.2/  ˛.2/ /

.h;˛/

D

X

h.1/ a.1/ ˝ .h.2/ / .˛.2/ /

by Lemma 4.3.4

.h;˛/

D

X

h.1/ .h.2/ /a.1/ ˝ .˛.2/ /

.h;˛/

D

X

ha.1/ ˝ .˛.2/ /

.˛/

Dh

X

a.1/ ˝ .˛.2/ /

.˛/

D h%.˛/: Next, we define a map ' W H ˝

R H

! H  by

'.h ˝ ˛/ D h  ˛: The map ' is an H -module homomorphism: '.k.h ˝ ˛// D '.kh ˝ ˛/ X D ˛.1/ ..kh//˛.2/ .˛/

D

X

˛.1/ ..h/.k//˛.2/

.˛/

D

X .˛/

Dk

˛.1/ ..h//˛.2/ ..k//˛.3/

X

˛.1/ ..h//˛.2/

.˛/

D k  .h  ˛/ D k  '.h ˝ ˛/: We show that % is an isomorphism of H -modules by showing that %' D IH ˝RH  R and that '% D IH  . Let h 2 H , ˛ 2 H  . We have %'.h ˝ ˛/ D %.h  ˛/ X D h.1/ ˝ .h.2/  ˛/ .h/

since ‰.˛/ D 1 ˝ ˛

4.3 Hopf Modules

79

D

X

h.1/ ˝ .h.2/ / .˛/

by Lemma 4.3.4

.h/

D

X

h.1/ .h.2/ / ˝ .˛/

.h/

D h ˝ .˛/ D h˝˛

since ˛ 2

R

H.

Moreover, 0 '%.˛/ D ' @

X

1 a.1/ ˝ .˛.2/ /A

.˛/

D

X

a.1/  ..a.2/ /  ˛.3/ /

.˛/

D

X

.a.1/ .a.2/ //  ˛.3/

.˛/

D

X

.a.1/ /˛.2/

.˛/

D ˛; and so the proof of the proposition is complete.

t u

We investigate some important consequences of Proposition 4.3.2. We employ the following lemma. Lemma 4.3.5. Let R be a PID, let F be a free R-module of rank m, and let M be a submodule of F . Then M is a free R-module of rank l  m. Proof. Our proof is by induction on m. Assume that m D 1, and let fx1 g be an R-basis for F . Let I be defined as I D fr 2 R W rx1 2 M g: Then I is an ideal of R, and since R is a PID, I D Ra for some a 2 R. If a D 0, then M D 0, and consequently M has rank 0. If a 6D 0, then M D R.ax1 /, and so M has rank 1. For the induction hypothesis, we assume that every submodule of a free Rmodule of rank m  1 is free and of rank l  m  1. Let F be a free R-module of rank m on the basis fx1 ; x2 ; : : : ; xm g. Let M be a submodule of F . Put N D M \ .Rx1 ˚ Rx2 ˚    ˚ Rxm1 /:

80

4 Hopf Algebras

Then N is a submodule of the free module Rx1 ˚   ˚ Rxm1 , and by the induction hypothesis N is free with rank  m  1. Let I D fr 2 R W x D r1 x1 C r2 x2 C    C rm1 xm1 C rxm 2 M for some r1 ; r2 ; : : : ; rm1 2 Rg: Then I is an ideal of R such that I D Ra for some a 2 R. If a D 0, then M is a submodule of the free rank m  1 R-module Rx1 ˚    ˚ Rxm1 , and so, by the induction hypothesis, M is free and of rank  m  1. If a 6D 0, let w be an element of M of the form w D r1 x1 C r2 x2 C    C rm1 xm1 C axm for some r1 ; r2 ; : : : ; rm1 2 R, and let x D s1 x1 C s2 x2 C    C sm1 xm1 C rxm ; for s1 ; s2 ; : : : ; sm1 ; r 2 R, be an element of M . Then there exists c 2 R for which x  cw 2 N ; thus M D N C Rw: Evidently, this sum is isomorphic to the direct sum of R-modules R ˚ R ˚    ˚ R; where the number of summands l satsifies l  m. We conclude that a submodule M of a free R-module of rank m is free and of rank l  m. t u Proposition 4.3.3. Let R be a PID, and let H be a cocommutative R-Hopf algebra R for which that is free and of rank n over R. Then there exists an integral ƒ 2 H R D Rƒ. H R Proof. Since R is a PID and H is a free R-module of rank n, the submodule H is free and of rank m, m  n over R. By Proposition 4.3.2, m  rank.H  / D rank.H / D n; R and so m D 1 since rank.H  / D rank.H /. This says thatR H is a free rank one R R-module, and so there exists an integral ƒ 2 H for which H D Rƒ. t u R An integral ƒ for which Rƒ D H is a generating integral for H . Proposition 4.3.4. Let R be a PID, and let H be a cocommutative R-Hopf algebra that is free and of rank n over R. Then H Š H  as H -modules.

4.3 Hopf Modules

81

Proof. By Proposition 4.3.3, there exists a generating integral ƒ for map  W H ! H  by .h/ D h  ƒ D

X

R

H.

Define a

ƒ.1/ ..h//ƒ.2/ :

.ƒ /

Then .hk/ D

X

ƒ.1/ ..hk//ƒ.2/

.ƒ /

D

X

ƒ.1/ ..k/.h//ƒ.2/

.ƒ /

D

X

ƒ.1/ ..k//ƒ.2/ ..h//ƒ.3/

.ƒ /

D

X

ƒ.1/ ..k//.h  ƒ.2/ /

.ƒ /

D h  .ƒ.1/ ..k//ƒ.2/ / D h  .k/; R so that  is H -linear. We claim that  is surjective. To this end, let ' W H ˝ H ! R H  be the map defined in the proof of Proposition 4.3.2. For h 2 H , ˛ 2 H  , '.h ˝ ˛/ D '.h ˝ rƒ / D r'.h ˝ ƒ / D r.h  ƒ / D r.h/: Let ˇ 2 H  . Since ' is surjective, there exists an element h ˝ rƒ 2 H ˝ which '.h ˝ rƒ / D ˇ. Now,

(4.9) R H

for

.rh/ D r.h/ D '.h ˝ rƒ /

by (4.9)

D ˇ; and so  is surjective. Since H and H  have the same dimension,  is bijective. Thus  is an isomorphism of H -modules. t u Even if R is not a PID, we still have the following important corollary. Corollary 4.3.1. H  is a locally free rank one H -module. Proof. Let P denote a prime ideal of R, and let RP denote the localization of R at P . Let HP D RP ˝R H . Then RP is a PID by Corollary 4.2.2, and so HP Š .HP / as HP -modules by Proposition 4.3.4. t u

82

4 Hopf Algebras

4.4 Hopf Orders Let R be an integral domain, let K be its field of fractions, and suppose that G is a finite Abelian group of order l. Then the group ring KG is a K-Hopf algebra with comultiplication KG W KG ! KG ˝K KG defined by g 7! g ˝ g, counit KG W KG ! K defined by g 7! 1, and coinverse KG W KG ! KG given by g 7! g 1 , for g 2 G. Definition 4.4.1. An R-order in KG is an R-submodule A of KG (a submodule of KG as an R-module) that satisfies the conditions (i) A is finitely generated as an R-module, (ii) A is closed under the multiplication of KG, and (iii) KA D KG. Proposition 4.4.1. Let R be a Dedekind domain, and let A be an R-order in KG. Then A is projective as an R-module. Proof. Since A  KG and KG is a torsion-free R-module, A is torsion-free. Thus, by [Rot02, Corollary 11.107], A is projective as an R-module. t u Definition 4.4.2. An R-order H in KG for which KG .H /  H ˝H is an R-Hopf order in KG. For example, the integral group ring RG is an R-Hopf order in KG. When G D 1, then the Hopf order R1 D R in K1 D K is the trivial Hopf order. Proposition 4.4.2. Let R be a local Dedekind domain, and let H be an R-Hopf order in KG. Then H is free over R of rank jGj. Proof. By Proposition 4.4.1, H is projective. Since H is also finitely generated, H is free of rank, say m [Ei95, p. 727, Exercise 4.11]. Moreover, m D jGj since KH D KG. t u Proposition 4.4.3. Let R be a Dedekind domain and H be an R-Hopf order in KG. Then RG  H . Proof. Let P be a non-zero prime ideal of R, and let RP denote the localization. Put HP D RP ˝R H . We show that RPG  HP . By Proposition 4.4.2, HP is free and of rank l D jGj. Let fb1 ; b2 ; b3 ; : : : ; bl g be an RP -basis for HP . Let g 2 G. We show that RPG  HP . Put I D fr 2 RP W RG 2 HP g: P Since KH P D KG, g D li D1 ui bi for elements ui 2 K. If each ui 2 Rp , then RPG  HP . Otherwise, let u01 ; u02 ; : : : ; u0k , 1  k  l, denote the subset of fui gli D1 for which 0 ui 2 KnRP , and let bi0 ; b20 ; : : : ; bk0 denote the corresponding basis elements. Then

4.4 Hopf Orders

83

g D hC

k X

u0i bi0

i D1

for some h 2 HP . Let vi D .u0i /1 . Note that vi 2 RP for 1  i  k. Let SD

k \

RP v i :

i D1

We claim that S D I . We first show that I  S . To this end, let r 2 RP be such that RG 2 HP . Then rh C r

k X

u0i bi0 D rh C

i D1

k X

ru0i bi0 2 HP ;

i D1

and hence ru0i 2 RP for all i . Put si D ru0i . Then r D si vi for i D 1; : : : ; k with si 2 RP . It follows that r 2 S . Next, let r 2 S . Then r D si vi for elements si 2 RP , i D 1; : : : ; k. Now rg D rh C r

k X

u0i bi0

i D1

D rh C

k X

ru0i bi0

i D1

D rh C

k X

.si vi /u0i bi0

i D1

D rh C

k X

si bi0 2 HP ;

i D1

and so r 2 I . It follows that I D S . Next, let J D fr 2 RP W rg ˝ g  HP ˝RP HP g; and let T D

k \

RP vi vj :

i;j D1

Then, by an argument analogous to that above, we have J D T . Now, by Corollary 4.2.1, vi D ai ni , where is a uniformizing parameter for RP , ai is a unit of RP , and ni  0 is an integer. Let n D minfni g. Then S is a principal ideal of RP of the form . n /, and T is a principal ideal of the form T D . 2n /. Thus S 2 D T and so I 2 D J . Observe that

84

4 Hopf Algebras

Ig ˝ g D KG .Ig/  KG .HP /  HP ˝RP HP ; and so I  J D I 2 . Thus, I D RP , RP G  HP , and consequently RP G  HP . Thus, \ \ RP G  HP : P prime

P prime

Now, by [CF67, Chapter I, 3, Lemma 1], RG D

\

RP G and H D

P prime

thus RG  H .

\ P prime

HP ; t u

Of course, an R-Hopf order H in the Hopf algebra KG is itself an R-Hopf algebra. Proposition 4.4.4. Let H be an R-Hopf order in KG, where G is a finite Abelian group of order l. Then H is an R-Hopf algebra. Proof. Clearly, H is a ring. Let KG W K ! KG denote the structure map of the K-algebra KG. Since H is an R-submodule of KG, KG restricted to H yields the R-module structure of H; hence H is an R-algebra. Let jH , jH , and jH denote the restrictions of KG , KG , and KG to H , respectively. Since KG .H /  H ˝R H , jH serves as a comultiplication map of H , and the coassociativity of KG guarantees that jH is coassociative. Moreover, KG .H /  R, and so jH is a counit map for H , and the counit property of KG implies that jH satisfies the counit property. .q1/ .q1/ .q1/ For an integer q  2, define Œq D mKG KG , where the maps mKG and .q1/ KG are defined as in 3.4. Then, for all g 2 G, KG .g/ D Œl  1.g/ D gl1 D g 1 : Thus

.l1/

KG .H / D Œl  1.H / D mKG

  .l1/ KG .H /  H

since mKG .H ˝R H /  H and KG .H /  H ˝R H . It follows that jH is a coinverse map for H , and the coinverse property of KG implies that H satisfies the coinverse property. Thus H is an R-Hopf algebra. t u As a Hopf algebra, the R-Hopf order H has an ideal of integrals that can be characterized as follows. R Proposition 4.4.5. Let H be the ideal of integrals of the Hopf order H in KG. R R P 1 Then H D H . H /e0 , where e0 D jGj g2G g. R Proof. Since H is finitely generated, so isRthe submodule H by Lemma 4.2.2. Let fƒ1 ; ƒ2 ; : : : ; ƒl g be a generating set for H . Now each ƒi is an integral of KG, and R so there exist elementsRki 2 K for which ƒi D ki e0 (by Proposition 4.1.4, KG D Ke0 ). It follows R that HR D Je0 , where J is the R-submodule of K generated by fki g. Hence H . H /DKG . H /DKG .Je0 /DJ , which proves the proposition. u t

4.4 Hopf Orders

85

Is there a notion of short exact sequence for Hopf orders in KG? Let G be a finite Abelian group, and let J be a subgroup of G. The sequence s

1!J !G!G!1 is a short exact sequence of groups where s is the canonical surjection G ! G D G=J , given as s.g/ D g C J . There is a short exact sequence of K-modules 0 ! .KJ/C KG ! KG ! KG ! 0 that passes to the short exact sequence of K-Hopf algebras s

K ! KJ ! KG ! KG ! KI see (4.2). Now, let H be a Hopf order in KG. Then H 00 D s.H / is an R-Hopf order in KG, and H 0 D H \KJ is an R-Hopf order in KJ. (Prove these facts as exercises.) We define the sequence R ! H 0 ! H ! H 00 ! R

(4.10)

to be a short exact sequence of R-Hopf orders. In general, for a fixed finite Abelian group G, there are many Hopf orders in the group ring KG, and classifying Hopf orders is an active area of research. For the remainder of this section, we assume the following conditions. For an integer n  1 and prime p 2 Z, let R be a local Dedekind domain containing a primitive pn th root of unity, which we denote as pn . Let K be the field of fractions of R. Let G D Cpn denote the cyclic group of order p n with hgi D Cpn , and let ij GO D CO pn be the character group generated by  with  i .g j / D h i ; g j i D pn . The n minimal idempotents of KG are fei g, 0  i  p  1, where p n 1

ei D

X

pin m g m :

mD0

The set fei g is a K-basis for KG. Let tr W KG ! K denote the trace map given as tr.x/ D

p n 1

X

 i .x/, x 2 KG.

i D0

The trace map determines a non-degenerate, symmetric, K-bilinear form on KG defined as B.x; y/ D tr.xy/: Let H be an R-Hopf order in KG. By Proposition 4.4.2, H is a free R-module of rank p n .

86

4 Hopf Algebras

The dual module of H is the R-module defined as H D D fx 2 KG W B.x; H /  Rg: In fact, H D is free over R of rank p n . Proposition 4.4.6. Suppose fb1 ; b2 ; : : : ; bpn g is an R-basis for H . There exists a basis fˇ1 ; ˇ2 ; : : : ; ˇpn g for H D , “the dual basis,” that satisfies B.ˇi ; bj / D ıij . Proof. Since KH D KG, fbi g is also a basis for KG. Let ff1 ; f2 : : : ; fpn g be the dual basis for KG satisfying fi .bj / D ıij . The bilinear form B has a unique representation K X Ai;j Bi;j ; BD i;j D1

where ai;j D B.bi ; bj / and where Bi;j are bilinear forms defined as Bi;j .˛; ˇ/ D fi .˛/fj .ˇ/ for ˛; ˇ 2 KG. pn Since B is non-degenerate, fB.; bj /gj D1 is a basis for KG . Consequently, the p n  p n matrix A D .B.bi ; bj // is invertible. Put A1 D .i;j /, and set ˇq D q;1 b1 C q;2 b2 C    C q;pn bpn for 1  q  p n . Then fˇ1 ; ˇ2 ; : : : ; ˇpn g is a basis for M D with n

B.ˇq ; bl / D

p X

ai;j Bi;j .ˇq ; bl /

i;j D1 n

D

p X

ai;l fi .ˇq /

i D1 n

D

p X

ai;l q;i D ılq :

t u

i D1

Proposition 4.4.7. Let H be an R-Hopf order in KG. Then .H D /D D H . t u

Proof. Exercise.

As an example of a dual module, we let H D RG. Then H has R-basis n f1; g; g 2 ; : : : ; gp 1 g. Since B.ei ; gj / D tr.ei g j / D ıij , the dual basis for RGD is fe0 ; e1 ; : : : ; epn 1 g. Consequently, RGD Š Re0 ˚ Re1 ˚    ˚ Repn 1 : Since KG Š Ke0 ˚ Ke1 ˚    ˚ Kepn 1 , RGD is the maximal R-order in KG.

4.4 Hopf Orders

87

Not surprisingly, H D can be identified with H  . Let x D Ppn 1 and let y D j D0 bj ej , bj 2 K, be an element of H . Then 0

pn 1

B.x; y/ D B @

X

ai gi ;

i D0

i D0

ai gi , ai 2 K,

1

p n 1

X

Ppn 1

bj ej A

j D0

p n 1

D

X

ai bj B.gi ; ej /

i;j D0 p n 1

D

X

ai bj tr.gi ej /

i;j D0 p n 1

D

X

ij

ai bj pn

i;j D0 p n 1

D

X

ai bj  i .g j /

i;j D0

D

*pn 1 X

p n 1 i

ai  ;

i D0

X

+ bj ej ;

j D0

and so H D can be identified with the set H  D fx 2 K CO pn W hx; H i  Rg: Indeed, H D Š H  as R-modules; the isomorphism is given by g i 7!  i for i D 0; : : : ; p n  1. In what follows, we make the identification H D D H  . Proposition 4.4.8. Suppose pn 2 K, and let H be an R-Hopf order in KG. Then H D is an R-Hopf order in KG. Proof. By definition, H D is an R-submodule of KG and is free over R of finite rank by Proposition 4.4.6. Since KG W H ! H ˝R H , KG W H  ˝R H  ! H  , and so H D D H  is closed under the multiplication of KG. Moreover, KH D D KG, and so H D is an R-order in KG. So it remains to show that KG.H D /  H D ˝ H D . But this follows from the fact that H is closed under the multiplication of KG. Thus H D is an R-Hopf order in KG. t u Proposition 4.4.9. Let H be an R-Hopf order in KG. Then there is an embedding pn 1 H ! RGD given by g 7! .1; pn ; p2 n ; : : : ; pn /.

88

4 Hopf Algebras

Proof. By Proposition 4.4.8, H D is a Hopf order in KG. By Proposition 4.4.3, there is an inclusion  W RG ! H D , and so there is a dual map  W .H D /D D H ! RGD , given as  .g/.f / D g..f // D g.f /. Noting that p n 1

g D e0 C pn e1 C p2 n e2 C    C pn

epn 1 t u

completes the proof.

One can show that an R-order H in KG is an R-Hopf order without explicitly showing that H is invariant under KG . We employ the discriminant of an R-order H in KG. Let fb1 ; b2 ; : : : ; bpn g be an R-basis for H . Then the discriminant of H with respect to the bilinear form B.x; y/ D tr.xy/ is defined as disc.H / D Rdet.A/; where A is the pn  p n matrix with ij th entry B.bi ; bj /. Proposition 4.4.10. Let n  1 be an integer, and suppose K contains a primitive O Suppose p n th root of unity. Let G be cyclic of order p n with character group G. H and J are R-orders in KG. If B.J; H /  R and disc.H D / D disc.J /, then J D H D and both H and J are Hopf orders in KG. Proof. Since hJ; H i  R, J  H D , and so, by [CF67, Chapter I, 3, Corollary 1], J D H D . Since J is an order over R in KG, J is an R-algebra with operations induced from KG. Consequently, comultiplication on KG restricts to give a comultiplication on J D D H , and so H is a Hopf order. It follows that H D D J is also a Hopf order. t u O The Let H be an R-Hopf order in KG. Then H  is an R-Hopf order in K G. integrals of H and H  are related by the following proposition due to R. Larson. R Proposition 4.4.11. Let H be an R-Hopf order in KG with integrals H , and let R H  be its linear dual, which is an R-Hopf order in K GO with integrals H  . Then R R H . H /H  . H  / D p n R. Proof. By Corollary 4.2.2, R is a PID, and so, by Proposition 4.3.3 and Proposition 4.3.4, there exists a generating integral ƒ for H  and an H -module isomorphism  W H ! H P with .h/.a/ D .ƒ / ƒ.1/ ..h//ƒ.2/ .a/ for h; a 2 H . Let h 2 H be such that .h/ D 1. Then, for a 2 H , .h/.a/ D 1.a/ D a.1/ D H .a/:

(4.11)

4.4 Hopf Orders

89

We claim that H .h/ is an integral of H . We have, for a; b 2 H , .H .b/h/.a/ D

X

ƒ.1/ .H .h/b/ƒ.2/ .a/

.ƒ /

D

X

ƒ.1/ .H .h//ƒ.2/ .b/ƒ.3/ .a/

.ƒ /

0 D@

X

1 ƒ.1/ .H .h/A ƒ.2/ .ba/

.ƒ /

D .h/.ba/ D H .b/H .a/

by (4.11)

D H .b/.h/.a/ D .H .b/h/.a/; and so .H .b/h/ D .H .b/h/. Since  is 1-1, H .b/h D H .b/h, and so H .h/b D H .H .b/h/ D H .H .b/h/ D H .b/H .h/: Thus, H .h/ 2

R

H.

By the definition of ,

1 D .h/.1/ D

X

ƒ.1/ .H .h//ƒ.2/ .1/ D H .h/.ƒ /:

(4.12)

.ƒ /

Set ƒ D H .h/. Since ƒ is an integral for KG, and since KG has a generating Ppn 1 integral p n e0 D i D0 g i , we have ƒ D qe0 for some q 2 K. Moreover, ƒ 2 R Ppn 1 i 1  O i D0  is a generating integral for K G. K GO , and so ƒ D t0 , since 0 D p n Now, by (4.12), 1 D ƒ.ƒ / D qtp n he0 ; 0 i D qt; and so H .ƒ/H  .ƒ / D qp n H .e0 /tH  .0 / D qtp n D pn ; which yields H .

R

H /H



.

R

H/

D p n R.

t u

We will presently give another useful formula. Let J be a subgroup of G, and consider the short exact sequence of K-Hopf algebras i

s

K ! KJ ! KG ! KM ! K;

M D G=J:

90

4 Hopf Algebras

Let H be an R-Hopf order in KG. Then, recalling (4.10), there is a short exact sequence of Hopf orders R ! H 0 ! H ! H 00 ! R: The sequence 



s i K ! KM  D K MO ! KG D K GO ! KJ  D K JO ! K

is a short exact sequence of K-Hopf algebras, which yields the short exact sequence of R-Hopf orders R ! .H 00 / ! H  ! .H 0 / ! R;

(4.13)

which is dual to (4.10). Proposition 4.4.12. With the notation above, R R R H . H / D H 0 . H 0 /H 00 . H 00 /. Proof. Let i

s

R ! H 0 ! H ! H 00 ! R be a short exact sequence of Hopf orders as in (4.10). By Proposition 4.3.3, H 0 has generating integral ƒ0 , H has generating integral ƒ, and H 00 has a generating integral ƒ00 . R Let ‚ be such that s.‚/ D ƒ00 . We claim that i.ƒ0 /‚ 2 H . Let a 2 H . Then s.a‚/ D s.a/ƒ00 D H 00 .s.a//ƒ00 D s.H .a//ƒ00 D H .a/ƒ00 ; and so a‚ D H .a/‚ C k, where k 2 i.H 0 C /H . Thus ai.ƒ0 /‚ D H .a/i.ƒ0 /‚ C ki.ƒ0 /; where ki.ƒ0 / 2 i.H 0 CR/i.ƒ0 /H D 0 since bƒ0 D H 0 .b/ƒ0 D 0 for b 2 H 0 C . It follows that i.ƒ0 /‚ 2 H . Now, i.ƒ0 /‚ D rƒ for some r 2 R, and so rH .ƒ/ D H .i.ƒ0 //H .‚/ D i.H 0 .ƒ0 //s.H .‚// D H 0 .ƒ0 /H 00 .ƒ00 /;

4.4 Hopf Orders

91

and so H 0

R  H0

H 00

R



H 00

 H

R  H .

A similar argument using the short exact sequence (4.13) shows that R  R  R  .H 00 / .H 00 / .H 0 / .H 0 / D t.H / .H / for t 2 R. Now, by Proposition 4.4.11, R  R 1 .H 0 / .H 0 / D .dim.H 0 //H 0 H 0 , R  R 1 .H 00 / .H 00 / D .dim.H 00 //H 00 H 00 , and .H / and so .tp n /H

R 1 H

R



.H /

D .p n /H

R 1 H

D .dim.H 0 //.dim.H 00 //H 0

Thus, H

R  H

 H 0

R  H0

H 00

,

R 1 H0

R H 00

H 00

R H 00

1

.

 , t u

which completes the proof of the proposition. We also have the following. Proposition 4.4.13. Let H be an R-Hopf order in KG. Then R pn disc.H / D H H , where disc.H / is defined with respect to the bilinear form B.x; y/ D tr.xy/.

Proof. Let  W H ! H  be the map defined as .h/ D h  ƒ , where ƒ is a generating integral for H  . Let fh1 ; h2 ; : : : ; hpn g be a basis for H , and let f˛1 ; ˛2 ; : : : ; ˛pn g be the dual basis for H  . We have ˛i .bj / D ıij . For i D 1; : : : ; p n , let ki 2 H be such that .ki / D ˛i . Then fk1 ; k2 ; : : : ; kpn g is a basis for H with .ki /.hj / D ˛i .hj / D ıij : Ppn For each l, 1  l  p n , we have hl D i D1 ri;l ki for some ri;l 2 R. Now 0 .hl /.hj / D  @

n

p X

1 ri;l ki A .hj /

i D1 n

D

p X i D1

ri;l .ki /.hj /

92

4 Hopf Algebras n

p X

D

ri;l ıij

i D1

D rj;l : Observe that the p n  p n matrix .rj;l / is invertible. Ppn 1 Pp n Since aD0  a is an integral of H  , bƒ D aD0  a for some b 2 R. Now, rj;l D .hl /.hj / D .hl  ƒ /.hj / 0 1 p n 1 X 1  a A .hj / D @hl  b aD0 0

1

p n 1

D

X

1 @ hl   a A .hj / b aD0

p 1 1 X a  .hl hj / D b aD0 n

D

1 tr.hl hj /; b

which yields brj;l D tr.hl hj /. Thus n

b p det..rj;l //R D disc.H /; and so

n

disc.H / D b p R; since det..rj;l // is a unit of R. R R  R  Now, by Proposition 4.4.5, R H D H H e0 , and since R is a PID, H H D cR for some c 2 R. Hence HRD cRe0 .R  By Proposition 4.4.11, H H H  H  D p n R, and so H .cRe0 /H  .b 1 p n R0 / D p n R; which yields cR D bR. Consequently, n

disc.H / D c p R D H

R pn H

. t u

4.5 Chapter Exercises

93

4.5 Chapter Exercises Exercises for 4.1 1. Let  W H ! R be the counit map of the R-Hopf algebra H . Let  W R ! H denote the R-algebra structure map. Show that the sequence 

id



R!H !H !R!R is short exact. 2. Let A, B be R-Hopf algebras. Verify that the maps .A ˝ B / W A ˝R B ! R ˝R R WD R and .A ˝ B / W A ˝R B ! A ˝R B; defined as .A ˝ B /.a ˝ b/ D A .a/B .b/ and .A ˝ B /.a ˝ b/ D A .a/ ˝ B .b/, respectively, satisfy the counit and coinverse properties, respectively. 3. Let A be an R-Hopf algebra. Is the comultiplication  W A ! A ˝R A a homomorphism of R-Hopf algebras? 4. Let W E ! F be an epimorphism of R-group schemes with RŒE D A, RŒF  D B. Prove that  W B ! A is a Hopf injection. 5. Prove that  i s  R ! RŒY; Y 1  ! RŒX; X 1 ! RCp ! R is a short exact sequence of R-Hopf algebras. Here  is the R-algebra structure map, i is the inclusion given by Y 7! X p , s is the quotient map with kernel .X p  1/, and  is the counit map of the R-Hopf algebra RCp . Exercises for 4.2 6. 7. 8. 9.

Prove that a PID is a Dedekind domain. Prove that ZŒi is a Dedekind domain. Give an example of a Noetherian integral domain that is not a Dedekind domain. Let K be a finite extension of Q with ring of integers R. Let J be a non-trivial proper ideal of R. Show that J can be factored into a product of prime ideals of R J D P1e1 P2e2    Pkek ; where ei are positive integers, by completing the following steps in the proof. (a) Show that J is contained in a prime ideal P1 of R. (b) ShowPthat there exist elements fqi g in Q and elements fmi g in J for which 1 D li D1 qi mi . (c) Show that there exists an ideal J1 of R for which J D J1 P1 . (d) Repeat the process with J1 in the role of J to obtain J D J2 P2 P1 for some ideal J2 and some prime ideal P2 .

94

4 Hopf Algebras

(e) Show that this process terminates so that J D Pl Pl1    P1 , where each ideal Pi is prime, 1  i  l. 10. Show that the factorization in Exercise 9 is unique up to a reordering of the factors Piei . Exercise for 4.3 11. Suppose p 2 R, and let Cp denote the cyclic group of order p. Then RCp is an RCp -module with scalar multiplication defined as .h  ˇ/.k/ D P p1  i .ˇ/ ˇ.1/ .RCp .h//ˇ.2/ .k/ for h; k 2 RCp , ˇ 2 RCp . Prove that fg  0 gi D0 P p1 ij is an R-basis for RCp , where 0 D p1 j D0  j  i .g j / D p . Exercises for 4.4 12. Let R be an integral domain that is a local ring with maximal ideal M . Let K be the field of fractions of R. Let n  1 be an integer, and assume that pn 2 K. Suppose that H is an R-Hopf order in KCpn for which H 6D RCpn . Prove that H is a local ring with maximal ideal .M; H C /. 13. Let s K ! KJ ! KG ! KG ! K be the short exact sequence of K-Hopf algebras given in 4.4. Let H be an R-Hopf order in KG. Prove that H 00 D s.H / is an R-Hopf order in KG and that H 0 D H \ KJ is an R-Hopf order in KJ.

Chapter 5

Valuations and Larson Orders

Throughout this chapter, G is a finite Abelian group and K is a finite extension of Q. R. Larson [Lar76] has constructed a collection of Hopf orders in KG that are known as Larson orders. At the core of Larson’s construction is the notion of valuations on K.

5.1 Valuations Definition 5.1.1. An absolute value on K is a function j j W K ! R that satisfies, for all a; b 2 K, (i) jaj  0, with jaj D 0 if and only if a D 0; (ii) jabj D jajjbj; (iii) ja C bj  jaj C jbj. For example, if K D Q and p is a prime of Z, then the p-adic absolute value j jp on Q is an absolute value. (Prove this as an exercise.) Definition 5.1.2. Let L=K be a field extension, and suppose k k is an absolute value on L and j j is an absolute value on K. Then k k is an extension of j j if kxk D jxj for all x 2 K. Definition 5.1.3. Let j j and j j0 be absolute values on K. Then j j and j j0 are equivalent if, for all x 2 K, the relation jxj < 1 implies jxj0 < 1. Using Definition 5.1.3, the absolute values on a field can be partitioned into equivalence classes. For the case K D Q, one can easily show that j jp is not equivalent to j jq if p 6D q and that the ordinary absolute value, which we denote by j j1 , is not equivalent to j jp for all p. (Prove these facts as exercises.) In fact, we have the Following proposition.

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 5, © Springer Science+Business Media, LLC 2011

95

96

5 Valuations and Larson Orders

Proposition 5.1.1. (A. Ostrowski) An absolute value j j on Q is equivalent either to j jp for exactly one prime p or to j j1 . Proof. For a proof, see [FT91, Theorem 9].

t u

So, up to equivalence, the collection of absolute values on Q consists of fj j2 ; j j3 ; j j5 ; : : : ; j j1 g: Let j j denote an absolute value on the field K. Then the sequence fan g in K is j j-Cauchy if for each  > 0 there exists an integer N > 0 for which jam  an j <  whenever m; n  N . The completion of K with respect to j j is the smallest field extension Kj j of K in which each Cauchy sequence converges to an element of Kj j . Note that j j extends uniquely to an absolute value (also denoted by j j ) on the completion. The completion of Q with respect to j jp is the field of p-adic rationals Qp , and j jp extends uniquely to an absolute value j jp on Qp . The completion of Q with respect to j j1 is R, and j j1 extends (uniquely) to the ordinary absolute value j j1 on R. Let K D Qp .˛/ be a simple algebraic extension of Qp . Proposition 5.1.2. The absolute value j jp on Qp has a unique extension j jp to K. Moreover, K is complete with respect to j jp . Proof. For a proof, see [CF67, Chapter II, 10, Theorem].

t u

But what happens if the base field is not complete? Let K D Q.˛/ be a simple algebraic extension of Q, and let r.x/ be the irreducible polynomial of ˛ of degree n D deg.r.x// D ŒK W Q. Let p be a prime of Z, and suppose that Q is endowed with the absolute value j jp . In what follows, we show how to extend j jp to the larger field K. Proposition 5.1.3. There are at most n extensions of j jp to Q.˛/. Proof. Let  W Q.˛/ ! Qp ˝Q Q.˛/ be the map defined by a 7! 1 ˝ a for all a 2 Q.˛/. Let l Y qj .x/ r.x/ D j D1

be the factorization of r.x/ over Qp into distinct irreducible polynomials qj .x/ 2 Qp Œx. For each j , 1  j  l, let ˛j be a zero of qj .x/, and put Kj D Qp .˛j /. There exists a homomorphism of rings j W Qp ˝Q Q.˛/ ! Qp .˛j /;

5.1 Valuations

97

defined by s˝

n1 X

rm ˛ m 7!

mD0

n1 X

s.rm =1/˛jm :

mD0

For each j , 1  j  l, let j W Q.˛/ ! Qp .˛j / be the ring homomorphism defined as j D j . Observe that ker.j / is a proper ideal of the field Q.˛/, and so ker.j / D f0g. Thus j is an injection. Since Qp is complete, j jp extends uniquely to an absolute value j jp;j on Qp .˛j / by Proposition 5.1.2. Now, for each j , 1  j  l, we define an extension k kp;j of j jp to Q.˛/ by the rule kakp;j D jj .a/jp;j for all a 2 Q.˛/. Moreover, by [La84, Chapter XII, 3, Proposition 3.1], an arbitrary absolute value j j on Q.˛/ extending j jp is equivalent to k kp;j for some j . And by [La84, Chapter XII, 3, Proposition 3.2], each k kp;j represents a distinct equivalence class of absolute values on Q.˛/. Thus, the extensions of j jp to Q.˛/ consist of fk kp;1 ; k kp;2 ; : : : ; k kp;l g; with l  n D ŒQ.˛/ W Q.

t u

For example, we take K D Q.i/ and compute the extensions of j j5 to K. We form the completion Q5 . By Hensel’s Lemma, Q5 contains the primitive 4th root of unity, i, and so the polynomial x 2 C 1, which is irreducible over Q, factors as x 2 C 1 D .x  i/.x C i/ over Q5 . Thus there are injections 1 W Q.i/ ! Q5 ; i 7! iI 2 W Q.i/ ! Q5 ; i 7! i: j5;1

j5;2

D j D j j5 . Thus, there are two absolute values on K that In this case, j extend j j5 , defined for a C bi 2 Q.i/ as ka C bik5;1 D ja C bij5 and ka C bik5;2 D ja  bij5 : We can recover the absolute values given in Proposition 5.1.3 in another way. Let K be a finite extension of Q with ring of integers R, and let p be a prime of Z. Since R is a Dedekind domain, there is a unique factorization e

.p/ D pR D P1e1 P2e2    Pg g of .p/ into distinct prime ideals of R.

98

5 Valuations and Larson Orders

Proposition 5.1.4. Each prime factor Pi in the factorization of .p/ determines an absolute value on K that extends j jp . Moreover, if Pi 6D Pj , then the absolute value determined by Pi is not equivalent to the absolute value given by Pj . Proof. Let q D st , for s 2 R, t 2 R , be an element of K. Let v be the integer for which .s/  Piv , .s/ 6 PivC1 , and let w be such that .t/  Piw , .t/ 6 PiwC1 . Now define  vw p w=ei 1 1 Œqp;i D v=e D .vw/=e D : i p i p p 1=ei Then Œ p;i is an absolute value on K that extends j jp . Now suppose Pi 6D Pj . There exists an element s 2 R for which s 2 Pi , s 62 Pj . Thus ordPi .s/ D v > 0, while ordPj .s/ D 0. Thus Œsp;i < 1, Œsp;j D 1, and so, by Definition 5.1.3, Œ p;i 6 Œ p;j . t u The absolute value Œ p;i is discrete since the set flogN .Œxp;i / W x 2 K  g;

N D p 1=ei ;

is the additive subgroup Z of the real numbers. We shall write the completion KŒ p;i as KPi . The Pi -order of q, denoted by ordPi .q/, is defined to be the integer v  w in the definition of Œqp;i , and the function ordPi W K ! Z [ 1, q 7! ordPi .q/, is the discrete valuation of K corresponding to the prime ideal Pi . We will show that representatives of equivalence classes of absolute values on K extending j jp can be chosen to be discrete absolute values. We need a lemma. Lemma 5.1.1. Let j W Qp ˝Q Q.˛/ ! Qp .˛j /, 1 ˝ ˛ 7! ˛j , be the homomorphism of rings defined in the proof of Proposition 5.1.3. Then there is an isomorphism  W Qp ˝Q Q.˛/ !

l M

Qp .˛j /;

(5.1)

j D1

L defined as .x/ D lj D1 j .x/. P Proof. Let y D niD0 .qi ˝ ˇi / be an element of Qp ˝Q Q.˛/. Then y D h.1 ˝ ˛/, where h.x/ is a polynomial in Qp Œx. If r.x/ 2 QŒx is the irreducible polynomial for ˛, then r.1 ˝ ˛/ D 1 ˝ r.˛/ D 0. Now, if .y/ D 0, then qj .x/ divides h.x/ Q for all j . Thus, r.x/ D lj D1 qj .x/ divides h.x/, and so y D h.1 ˝ ˛/ D 0. Thus  is an injective ring homomorphism. L Now, as vector spaces over Qp , dim.Qp ˝Q Q.˛// D dim. lj D1 Qp .˛j //, and so  is surjective. Thus  is a ring isomorphism. t u

5.1 Valuations

99 g

Proposition 5.1.5. The collection of prime factors fPi gi D1 in the factorization of .p/ is in a 1-1 correspondence with the equivalence classes of absolute values on K that extend j jp . g

Proof. Define a map ˆ W fPi gi D1 ! fk kglj D1 , where ˆ.Pi / is the representative of the equivalence class containing the discrete absolute value Œ p;i . If ˆ.Pi / D ˆ.Pk / then Œ p;i is equivalent to Œ p;j , which can only happen if i D j . Thus, ˆ is an injection. L By Lemma 5.1.1, there is a ring isomorphism  W Qp ˝Q Q.˛/ ! lj D1 Qp .˛j /. Put 1 D 1 .1; 0; 0 : : : ; 0/; 2 D 1 .0; 1; 0; : : : ; 0/; : : : ; l D 1 .0; 0; 0; : : : ; 1/: Then fi g is the set of primitive idempotents of Qp ˝Q Q.˛/. By [FT91, Chapter III, 1, Theorem 17], the set fi gli D1 is in a 1-1 correspondence with the set of prime g ideals fPi gi D1 . Consequently, l D g, and so ˆ is a bijection. t u We have not discussed how the absolute value j j1 extends to a simple field extension K D Q.˛/. Proposition 5.1.6. Let n D ŒK W Q. There are at most n extensions of j j1 to K. Proof. The proof is similar to the proof of Proposition 5.1.3. Let  W Q.˛/ ! R ˝Q Q.˛/ be the map defined by a 7! 1 ˝ a for a 2 Q.˛/. Let r.x/ D

l Y

qj .x/

j D1

be the factorization of r.x/ over R into distinct irreducible polynomials qj .x/. For each j , 1  j  l, let ˛j be a zero of qj .x/ and put Kj D R.˛j /. Of course, either R.˛j / D R or R.˛j / D C. For each j , 1  j  l, there exists a homomorphism of rings j W R ˝Q Q.˛/ ! R.˛j /; defined by s˝

n1 X mD0

rm ˛ m 7!

n1 X

srm ˛jm ;

mD0

and an injection j W Q.˛/ ! R.˛j /; given as j D j . Since R is complete, j j1 extends uniquely to an absolute value on R.˛j / that is either the ordinary absolute value j j on R or C. Now, for each j , 1  j  l, we can define an extension k k1;j of j j1 to K D Q.˛/ as kak1;j D jj .a/j

100

5 Valuations and Larson Orders

for all a 2 Q.˛/. By [La84, Chapter XII, 3], the equivalence classes of extensions of j j1 to K have representatives fk k1;1 ; k k1;2 ; : : : ; k k1;l g; with l  n D ŒK W Q.

t u

An extension k k1;j of j j1 is an Archimedean absolute value. Analogous to (5.1), one has a ring isomorphism  W R ˝Q Q.˛/ Š

s M

R˚

i D1

t M

C;

(5.2)

j D1

L L defined as .x/ D siD1 i .x/ ˚ tj D1 j .x/, where i , 1  i  s, are the real embeddings and j , 1  j  t, are the complex embeddings.

5.2 Group Valuations Definition 5.2.1. Let G be a finite group (not necessarily Abelian!). A group valuation is a function  W G ! Z [ 1 that satisfies, for all g; h 2 G, (i) (ii) (iii) (iv)

.g/  0, .g/ D 1 if and only if g D 1, .gh/  minf.g/; .h/g, and .ghg1 h1 /  .g/ C .h/.

Let us review some basic properties of the group valuation  on G. Proposition 5.2.1. For all g; h 2 G, (i) .g/ D .g 1 / and (ii) .ghg1 / D .h/. Proof. To prove (i), let s D jhgij, and observe that g s1 D g 1 . Since g D .g s1 /s1 , Definition 5.2.1(iii) implies that .g/ D ..gs1 /s1 /  .gs1 / D .g 1 /: On the other hand, .g 1 / D .g s1 /  .g/. For (ii), one has, for all h; g 2 G, .ghg 1 / D .ghg1 h1 h/  minf.ghg 1 h1 /; .h/g  minf.g/ C .h/; .h/g  .h/:

by Def. 5.2.1(iii) by Def. 5.2.1(iv)

5.2 Group Valuations

101

Now, with k D ghg 1 , .h/ D .g1 kg/  .k/ D .ghg1 /  .h/; and thus .h/ D .ghg1 /.

t u

Let Z0 D fr 2 Z W r  0g, and let  be a group valuation on G. For each r 2 Z0 , put Gr D fg 2 G W .g/  rg: Proposition 5.2.2. For each r 2 Z0 , Gr is a normal subgroup of G. Proof. We first show that Gr is a subgroup of G. Since .1/ D 1 > r, 1 2 Gr . Let hr ; gr 2 Gr . Then .hr gr /  minf.hr /; .gr /g  r, and so Gr is closed under the 1 binary operation of G. Also, .h1 2 Gr . Thus Gr  G. r / D .hr /  r, and so h 1 Now, let ghr 2 gGr . Then .ghr g / D .hr /  r by Proposition 5.2.1(ii), and so ghr g 1 2 Gr . Thus gGr  Gr g. Next, let hr g 2 Gr g. Then .g1 hr g/ D .hr /  r, and so g 1 hr g 2 Gr . Thus, Gr g  gGr . t u Let  be a group valuation on G with jGj  2. Put m D maxf.g/ W g 2 G; g 6D 1g. Proposition 5.2.3. There is a normal series 1  Gm  Gm1      G1  G0 D G; which can be refined to a composition series 1  Ns  Ns1      N1  N0 D G: Proof. Note that G0 D G, 1 D GmC1 , and 1  Gm since there exists at least one g 2 G with 1 > .g/  m. Now, by Proposition 5.2.2, Gj G G for 0  j  m, and thus the series 1  Gm  Gm1      G1  G0 D G

(5.3)

is normal. We first remove all repeats from the series (5.3) and renumber the indices (if necessary) to obtain the normal series 1  Gn  Gn1      G1  G0 D G:

(5.4)

Let i be an index, 0  i  n  1, for which Gi C1  Gi . Let Qi D Gi =Gi C1 . If Qi has no proper non-trivial normal subgroups (that is, if Qi is simple), then set Ni D Gi and Ni C1 D Gi C1 , and replace Gi C1  Gi in the normal series (5.4) with Ni C1  Ni . If Qi has a proper non-trivial normal subgroup, say M 0 , then necessarily there exists a subgroup M of Gi with Gi C1  M  Gi . Set Ni 1 D Gi ,

102

5 Valuations and Larson Orders

Ni D M , and Ni C1 D Gi C1 , and replace Gi C1  Gi in the normal series (5.4) with Ni C1  Ni  Ni 1 . We now repeat this process with the proper containments Ni C1  Ni and Ni  Ni 1 , replacing these if necessary, with refinements as above. Eventually we obtain a normal series 1  Ns  Ns1      N1  N0 D G; which contains the normal series (5.4) and for which each quotient Nj =Nj C1 is simple, that is, we obtain the composition series for G. t u The properties given above help to determine the group valuations on a given group. For example, we classify all group valuations on D3 , the dihedral group of order 6, which has group presentation D3 D ha; b W a3 D 1; b 2 D 1; bab D a1 i: Let  W D3 ! Z [ 1 be a group valuation. Then .1/ D 1. The non-trivial conjugacy classes of D3 are fa; a2 g and fb; ab; a2 bg. Thus, by Proposition 5.2.1(ii), we have .a/ D .a2 / D r and .b/ D .ab/ D .a2 b/ D s for some r; s 2 Z0 . Moreover, since baba1 D a, we have r D .a/ D .baba1 /  .b/ C .a/ D s C r by Definition 5.2.1(iv). Thus r  s D 0. If r D s D 0, then we have the trivial group valuation on D3 : 0 .g/ D 0 for all g 2 D3 , g 6D 1. If r > 0 and s D 0, then we have the group valuation r with r .a/ D r .a2 / D r and r .b/ D r .ab/ D r .a2 b/ D 0. Thus 0 and r , r > 0, are the only group valuations on D3 . We consider r with r D 3. Then maxf3 .g/ W g 2 G; g 6D 1g D 3 and G0 D D3 , G1 D G2 D G3 D f1; a; a2 g, and G4 D 1, so the normal series is 1  f1; a; a2 g  D3 ; which is the composition series for D3 . In the case where G is a p-group, we have the following proposition. Proposition 5.2.4. Let  be a group valuation on G, where G is a p-group of order p n . Then  has at most n distinct finite values. Proof. Let 1  Gm  Gm1      G1  G0 D G be the normal series determined by , and let 1  Ns  Ns1      N1  N0 D G

5.2 Group Valuations

103

be the composition series. Then Ni =Ni C1 Š Cp for i D 0; 1; : : : ; s (here NsC1 D 1). Note that s D n  1. For each i , 0  i  s, there is a subseries GrC1  Ni C1  Ni  Gr : Let g 2 Ni nNi C1 . Since g 2 Gr , .g/  r. But since g 62 Ni C1 , g 62 GrC1 , and thus .g/ 6 r C 1, and so .g/ D r. Put ri D r, and let fri g denote the collection of the ri as i ranges from 0 to s. Since each g 2 G, g 6D 1, is in Ni nNi C1 for exactly one i , the collection fri gsiD0 constitutes the values of . At most n of the values in fri gsiD0 are distinct since s C 1 D n. t u Let K be a simple algebraic extension of Q. How do group valuations relate to discrete valuations on K? Let p be a fixed prime of Z, let P be a prime in the factorization of .p/, and let ordP W K ! Z [ 1 be the corresponding discrete valuation on K. For an ideal I of R, one has ordP .I / D v, where v is the integer for which I  P v , I 6 P vC1 . We set e D ordP ..p//. Definition 5.2.2. Let G be a finite group. The group valuation  W G ! Z [ 1 is order bounded with respect to ordP if (i) .G/  ordP .K/, (ii) .g/ D 0 for jgj not a power of p, and (iii) .g/  e=.p a  p a1 / for jgj D p a , a  1. An order-bounded group valuation (obgv)  on G is p-adic if .g p /  p.g/ for all g 2 G. Counting all of the p-adic order-bounded group valuations on a given group is an interesting problem. For example, let K D Q. p3 /, where p3 is a primitive p 3 rd 2 root of unity. The ring of integers of K is R D ZŒ p3 , and one has .p/ D P p .p1/ , with P D .1  p3 /. Let ordP denote the discrete valuation corresponding to P . Then e D ordP ..p// D p 2 .p  1/. Let Cp3 denote the cyclic group of order p 3 , and let  W Cp3 ! Z denote a p-adic obgv on Cp3 with respect to ordP . By Proposition 5.2.4,  has at most three distinct finite values fr0 ; r1 ; r2 g, satisfying ri C1  ri  0 for i D 0; 1. Since  is order bounded, e i 0  ri D .g p /  2i D pi ; p .p  1/ and so 0  r0  1; 0  r1  p; 0  r2  p 2 : Now, since  is a p-adic obgv on Cp3 , we have the additional condition r2  pr1  p 2 r0 ;

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5 Valuations and Larson Orders

and from this one finds that there are p 2 .p C 1/ CpC2 2 p-adic obgvs on Cp3 . For more results on counting group valuations, see the paper [KM07] by A. Koch and A. Malagon.

5.3 Larson Orders Let K be a finite extension of Q with ring of integers R, let p be a prime of Z, and let P be a prime ideal of R that lies above p; that is, P is in the prime factorization of p. Let ordP denote the corresponding discrete valuation on K. Then ordP .P / D 1 and ordP .P r / D r for an integer r  0. Let P r D .P r /1 D fx 2 K W xP r  Rg: Then, for y 2 P r , y 6D 0, yP r  R, and so P r  y 1 R. Since y 1 R is a finitely generated module over the Dedekind domain R, P r is finitely generated by Lemma 4.2.2. R. Larson [Lar76] has shown that order-bounded group valuations on G give rise to Hopf orders in KG. Proposition 5.3.1. Let p be prime, let P be a prime ideal of R that lies above p, let ordP be the corresponding valuation on K, let G be a finite Abelian group, and let  be a group valuation on G that is order bounded with respect to ordP . Let A./ be the R-subalgebra of KG generated by P  .g/ .g  1/ for g 6D 1. Then A./ is an R-Hopf order in KG. Proof. We sketch Larson’s proof. Let S denote the set of elements of G that have order p a for some a > 0. The elements of S can be ordered by the rule g  h if and only if .g/  .h/. We have the list of elements of S g1  g2  g3      gl : For each i , 1  i  l, let Agi be the R-subalgebra of Khgi i of the form Agi D RŒP  .gi / .gi  1/:

5.3 Larson Orders

105

Since P  .gi / is finitely generated, each Agi is a finitely generated R-module, and L thus the direct sum li D1 Agi is finitely generated. There exists a surjective map of R-modules l M Agi ! A./; sW i D1

which shows that A./ is finitely generated. One easily verifies the remaining conditions of Definition 4.4.1, and so A./ is an R-order in KG. Also, since KG .g/ D g ˝ g for all g 2 G,   g1 g1 g1 g1

KG D  .g/ ˝ 1 C 1 ˝  .g/ C .g  1/ ˝  .g/ ; P  .g/ P P P and so

KG .A.//  A./ ˝R A./: t u

Thus, A./ is an R-Hopf order in KG.

A Hopf order H in KG is a Larson order in KG if H is of the form A./, where  is a p-adic obgv. We consider Larson orders in KG where G D Cpn is the cyclic group of order p n generated by g. We consider the case n D 1. To characterize a Larson order in KCp , we need only describe the possible p-adic obgvs on Cp . By Proposition 5.2.4, an obgv  on Cp has exactly one finite value: we have .g/ D i for g 6D 1. Since  is order bounded, 0  i  e 0 , where e 0 D e=.p  1/. Since .g p / D .1/ D 1  .g/,  is p-adic. The Larson order H D A./ is written 

 g  1 g2  1 gp1  1 H D A./ D R ; ;:::; : Pi Pi Pi  Proposition 5.3.2. H D R

 g1 . Pi

i h  A./, so we prove the reverse containment. For Proof. Clearly, R g1 i P a D 2; 3; : : : ; p  1, one has ! ! ! a a a a a a1 a2 .g  1/ .g  1/ .g  1/; C CC g  1 D .g  1/ C 1 2 a1 and so ga  1  P i.a1/ Pi



g1 Pi

a

!  a1 a i.a2/ g  1 P C 1 Pi

!   a g1 Pi CC : Pi a1 Thus

g a 1 Pi

R

h

g1 Pi

i , which proves the proposition.

t u

106

5 Valuations and Larson Orders

We shall denote the Larson order of Proposition 5.3.2 by H.i / since it depends on only one valuation parameter i , 0  i  e0 . Proposition 5.3.3. Let H.i / be a Larson order in KCp . Let e D ordP ..p//. Then R ordP .H.i / . H.i / // D e  .p  1/i . domain Proof. We localize at the prime ideal P . Then RP is anh integral i g1 is written with K D Frac.RP /. Over RP , the Hopf order H.i / D R P i h i H.i /P D RP Pg1 . iR P By Corollary 4.2.2, RP is a PID, and so there h is ian element 2 RP for . Moreover, since H.i /P which . / D PRP . Consequently, H.i /P D RP g1 i is a finitely generated and projective module over the local ring RP , H.i /P is free over RP of rank p. Thus, (

    ) g  1 p1 g1 g1 2 ; ;:::; 1; i i i

(5.5)

is an RP -basis for H.i /P . With respect to this basis, we compute disc.H.i /P =RP /. Observe that the matrix 0 1 B0 B B B 0 M DB B B B @ 0

0 i

0 0

0

2i

0





0 0 :: : 0

0



1 C C C C C C C C A

.p1/i

multiplies the basis (5.5) to yield a basis for RP Cp . One has .det.M // D . /ip.p1/=2 , and so disc.RP Cp / D . /ip.p1/ disc.H.i /P =RP /: Now, since disc.RP Cp =RP / D .pRP /p D . /pe , . /pe D . /ip.p1/ disc.H.i /P =RP / or disc.H.i /P =RP / D . /pepi.p1/ D .. /ei.p1/ /p :

5.3 Larson Orders

107

R  Now, by Proposition 4.4.13, H.i /P H.i /P D . /ei.p1/ , which implies that  R  ordP H.i / H.i / D e  .p  1/i . t u We next consider Larson orders in KG D KCp2 . By Proposition 5.2.4, a group valuation  on Cp2 has at most two finite values: .ga / D i , for 1  a  p 2  1, a  0 .mod p/, and .g b / D j , for 1  b  p 2  1, b 6 0 .mod p/. Since  is order bounded, 0  i  e 0 and 0  j  e 0 =p, and since  is p-adic, pj  i . The Larson order H D A./ can be written  H D A./ D R

g a  1 gb  1 ; Pi Pj

 ;

where 1  a  p 2  1, a  0 .mod p/, 1  b  p 2  1, b 6 0 .mod p/. h p i g1 Proposition 5.3.4. H D A./ D R g P1 . i ; Pj Proof. The proof of Proposition 5.3.2 shows that  A./ D R h Certainly R

g p 1 g1 ; Pj Pi 2

i

gp  1 Pi

 

gb  1 Pj

 :

 A./, so it remains to show that A./  R

h

g p 1 g1 ; Pj Pi

i .

For b D 2; 3; : : : ; p  1, b 6 0 .mod p/, one has ! ! ! b b b .g  1/b1 C .g  1/b2 C  C .g  1/; gb  1 D .g  1/b C 1 2 b1 and so gb  1  P j.b1/ Pj



g1 Pj

b

!   b g  1 b1 P j.b2/ C 1 Pj

!   b g1 j CC P : b1 Pj Thus

g b 1 Pi

R

h

g p 1 g1 ; Pj Pi

i , which proves the proposition.

t u

We denote the Larson order H D A./ in KCp2 by H.i; j / since it depends on only two valuation parameters, i; j .

108

5 Valuations and Larson Orders

Proposition 5.3.5. Let H.i; j / be a Larson order in KCp2 . Then there exists a short exact sequence of Larson orders R ! H.i / ! H.i; j / ! H.j / ! R: Proof. There exists a short exact sequence of K-Hopf algebras s

K ! Khg p i ! KCp2 ! Khgi ! K; s.gp / D 1; h p i i h g 1 p i \ H.i; j / D H.i / D R and Khg . with s.H.i; j // D H.j / D R g1 j P Pi Thus the claimed short exact sequence exists. t u Proposition 5.3.6. Let H.i; j / be a Larson order in KCp2 . Then  R  ordP H.i;j / H.i;j / D 2e  .p  1/.i C j /. Proof. In view of the short exact sequence of Proposition 5.3.5, one has  R   R   R  ordP H.i;j / H.i;j / D ordP H.i / H.i / C ordP H.j / H.j /  R  by Proposition 4.4.12. Thus ordP H.i;j / H.i;j / D 2e  .p  1/.i C j / by Proposition 5.3.3. t u For the general case, Larson orders in KCpn can be described as follows. Proposition 5.3.7. Let n  1, and let  be a p-adic obgv on Cpn . Then the Larson order A./ in KCpn has the form " n1 # n2 gp  1 gp  1 gp  1 g  1 A./ D R ; ;:::; i ; ; P i1 P i2 P n1 P in where ir , 1  r  n, are integers satisfying 0  pir  ir1 for 2  r  n. Proof. By Proposition 5.2.4, a p-adic obgv on Cpn has n distinct finite values nr .g p / D ir , for r D 1;    ; n, that satisfy 0  pir  ir1 for 2  r  n. We proceed by induction on n. We have already seen that the proposition is true for the cases n D 1; 2, and so we assume that the .n  1/st case holds. Thus, 

 ga  1 n W 1  a  p  1; a  0 .mod p/ P  .ga /   b g 1 n W 1  b  p  1; b  6 0 .mod p/  P in # " n1 n2 gp  1 gp  1 gp  1 ; ;:::; i DR P i1 P i2 P n1   b g 1 n W 1  b  p  1; b  6 0 .mod p/ :  P in

A./ D R

5.3 Larson Orders

109

Since gb  1 P in .b1/ P in



g1 P in

b

!  b1 b in .b2/ g  1 C P 1 P in

!   b g1 CC P in ; b1 P in one concludes that " A./ D R

gp

n1

P

 1 gp ; i 1

n2

1

P i2

gp  1 ;:::; i P n1

#

 g1 : P in

t u

Larson orders in KCpn will be denoted as H.i1 ; i2 ; : : : ; in /. For a finite Abelian group G, not every Hopf order in KG is a Larson order, however. For example, if K D Q. p2 / and G D Cp2 is the cyclic group of order p 2 , then RG  is an R-Hopf order in K GO that is not Larson. Proposition 5.3.8. The R-Hopf order RCp2 is not of the form A./ for any p-adic obgv  on Cp2 . Proof. By way of contradiction, we assume that RCp2 D A./ for some p-adic obgv  on Cp2 . By Proposition 5.2.4, an obgv on Cp2 has at most two finite values: .gpa / D i for a D 1; 2; : : : ; p  1, and .gb / D j for .b; p/ D 1. Thus, 0  i  e=.p  1/ D p.p  1/=.p  1/ D p, and 0  j  e=p.p  1/ D 1. By Proposition 4.4.11, !! !! Z Z ordP RCp2 C ordP RC2 D 2e D 2p.p  1/; p

RCp 2

RC2 p

    R  R D 0. and so, since ordP RCp2 RC 2 D 2p.p  1/, ordP RC 2 RC  p

Now, by Proposition 5.3.6,  Z ordP A./

p

p2

 D 2p.p  1/  .p  1/.i C j /;

A./

and so 2p.p  1/ D .p  1/.i C j /. Thus, 2p D i C j . However, by the bounds t u on i; j , we have 1 C p  2p, which is impossible. Thus A./ 6D RCp2 . Every Hopf order in KG with G finite determines an obgv on G, however. We prove this fact below for the special case where G is Abelian of order m and m 2 K. Proposition 5.3.9. Let G be a finite Abelian group of order m, assume that m 2 K, and let H be an R-Hopf order in KG. Then H determines an order-bounded group valuation on G.

110

5 Valuations and Larson Orders

Proof. Let p be a rational prime, and let P be a prime of R lying above the prime p. Define .g/ D 1 if g D 1. For each g 2 G, g 6D 1, let jgj D k. If k is not a power of p, set .g/ D 0. If jgj is a power of p, then define Ig D fx 2 K W x.g  1/ 2 H g; Ig1

D fx 2 K W xIg 2 Rg:

By Proposition 4.4.3, RG  H , and so R  Ig . Thus Ig1 is an ideal of R. Put .g/ D ordP .Ig1 /. Thus .g/  0 for all g 2 G, and so the function  W G ! Z [ 1 satisfies Definition 5.2.1(i),(ii). For g; h 2 G, we have gh  1 D .g  1/h C h  1; and so Ig \ Ih  Igh . Thus, 1  .Ig \ Ih /1 D Ig1 C Ih1 : Igh

Consequently, .gh/  minf.g/; .h/g; which shows that  is a group valuation. To show that  is order bounded with respect to ordP , let g 2 G, and let Tg W KG ! KG denote the linear transformation defined as Tg .a/ D ag for a 2 KG. Put t D m=k, an integer. Then the characteristic polynomial of Tg is .x k  1/t . The eigenvalues of Tg are the kth roots of unity kl , l D 0;    ; k  1, each occurring with multiplicity t. For each l, there is a set of t linearly independent eigenvectors fal;j gtj D1 for kl , so that Tg .al;j / D al;j g D kl al;j . The set f kl al;j g for l D 0; : : : ; k  1, j D 1; : : : ; t, is a K-basis for KG. Now al;j Ig .g  1/ D Ig . kl  1/al;j : The Hopf order H is a finitely generated module over the Dedekind domain R, and thus the submodule Ig .g  1/  H is finitely generated. Consequently, Ig . kl  1/ is an integral ideal of R, and so R. kl  1/  Ig1 . Thus .g/ D ordP .Ig1 /  ordP .. kl  1//: Put k D p a . Observe that e D ordP ..p// D ordP .. kl  1/p Dp

a1

.p 

a1 .p1/

1/ordP . kl

/

 1/;

5.3 Larson Orders

so that

111

e D ordP . kl  1/: p a1 .p  1/

Thus .g/  ordP . kl  1/ D

e , p a1 .p1/

and so  is order bounded.

t u

With a little more work, one can show that  is a p-adic obgv on G. Thus each R-Hopf order H in KG gives rise to a p-adic obgv on G that we denote as „.H /. Given an R-Hopf order H in KG, „.H /, in turn, yields a Larson order A.„.H // that is contained in H . For a given H , A.„.H // is the largest Larson order contained in H . The relationship between A and „ is given as follows. Proposition 5.3.10. Let H be an R-Hopf order in KG. Let  be an obgv on G. Then: (i) A.„.H //  H , with equality holding if H is a Larson order. (ii)   „.A.//, with equality holding if  is p-adic. t u

Proof. Exercise.

We can consider Larson orders over complete local fields. Let K be a finite extension of Q, and let R denote the integral closure of Z in K. Let H.i1 ; i2 ; : : : ; in / denote a Larson order in KCpn . Let P be a prime ideal of R that lies above p. Let Πp be the discrete absolute value corresponding to P , and let KP denote the completion of K with respect to Πp . Note that Πp restricts to R and the localization RP . We denote the completion of RP with respect to Πp by RO P . Of course, the completion of R with respect to Πp is RO P . Proposition 5.3.11. The field KP is a finite extension of Qp . e

Proof. Let .p/ D Piei P2e2    Pg g denote the factorization of .p/ in R. By [FT91, Chapter III, 1, Theorem 17], there exists a ring isomorphism Qp ˝Q K !

g M

KPi :

i D1

of Q, KPi is a field Since Œ p;i is an extension of j jp and K is a field extensionL g extension of Qp . Thus KPi is a vector space over Qp , and so i D1 KPi is a vector space Lg over Qp . Since Qp ˝Q K is finite-dimensional over Qp , the same is true for t u i D1 KPi . Thus each KPi is finite-dimensional over Qp . Proposition 5.3.12. With the notation above, (i) the ring of integers in KP is the completion of R with respect to Œ p ; (ii) the completion of R with respect to Œ p (D RO P ) is a local Dedekind domain with maximal ideal P RO P ; P RO P being a principal ideal; and (iii) each element x 2 KP has a representation x D u r , where generates P RO P , r 2 Z, and u is a unit in RO P .

112

5 Valuations and Larson Orders

Proof. Let O denote the integral closure of RO P in KP . Suppose x D lim an 2 O. Then x satisfies a monic polynomial with coefficients in RO P , that is, .lim an /l D r0 C r1 .lim an / C r2 .lim an /2 C    C rl1 .lim an /l1 for ri 2 RO P . By continuity, lim anl D lim.r0 C r1 an C r2 an2 C    C rl1 anl1 /; and thus, for each  > 0, there exists an integer N for which Œanl  .r0 C r1 an C r2 an2 C    C rl1 anl1 /p <  for n  N . Consequently, there exists an integer M for which an 2 RP for all n  M . Therefore, fan g is a Œ p -Cauchy sequence in RP , and so x 2 RO P . This shows that O  RO P . Clearly, RO P  O, which proves (i). To prove (ii), note that RP is a local Dedekind domain with maximal ideal . / D PRP , and thus RO P is a local Dedekind domain with maximal ideal .lim /RO P D RO P . We leave (iii) as an exercise. t u Observe that the completion of Q with respect to j jp is the field of p-adic rationals Qp , and the ring of integers in Qp is the completion of Z with respect to j jp , the ring of p-adic integers, ZO .p/ . In what follows, we shall use the simpler notation Zp to denote the p-adic integers. By Proposition 5.3.12(iii), the discrete valuation ordP W K ! Z[1 extends to a discrete valuation on KP , which we denote by ord. For x 2 KP , one has ord.x/ D r, where x D u r . Let H.i1 ; : : : ; in / be a Larson order in KCpn . We complete R with respect to Œ p , and let H.i1 ; : : : ; in /P D RO P ˝P H.i1 ; : : : ; in /: Then

" H.i1 ; : : : ; in /P D RO P

gp

n1

 1 gp ; i

1

n2

1

i2

# g1 ;:::; i ; n

where P RO P D . /, and H.i1 ; : : : ; in /P is an RO P -Hopf order   in KP Cpn . g  1 , and Larson orders in Thus Larson orders in KP Cp appear as RO p i1  p  g 1 g1 ; i , for integers i1 ; i2 with 0  i1 ; i2  e 0 , KP Cp2 appear as RO p i1 2 pi2  i1 .

5.4 Chapter Exercises

113

5.4 Chapter Exercises Exercises for 5.1 1. Show that the map j jp W Q ! R defined as ˇa ˇ 1 ˇ rˇ ˇ p ˇ D r; b p p where .a; p/ D 1, .b; p/ D 1, and r 2 Z, is an absolute value on Q. 2. Prove that j jp is not equivalent to j jq if p 6D q, and that j j1 is not equivalent to j jp for all p. Exercises for 5.2 3. Classify all of the group valuations on D4 , the fourth order dihedral group. 4. Classify all of the p-adic order-bounded group valuations (obgvs) on D3 , the third-order dihedral group. 5. Let n  1. Compute the number of p-adic obgvs on Cpn when K D Q. pn /. 6. Let n  1. Compute the number of p-adic obgvs on Cpn when K D Q. pn /. 7. Let ordP be a discrete valuation on K D Q.˛/. Prove the following. (a) ordP W K ! Z [ 1 is a surjective group homomorphism. (b) ordP .xy/ D ordP .x/ C ordP .y/ for all x; y 2 K. (c) ordP .x C y/  minfordP .x/; ordP .y/g for all x; y 2 K. 8. Prove that x 2 C 1 factors over Qp if and only if p  1 .mod 4/. Exercises for 5.3 9. Prove that every obgv on Cp is a p-adic obgv. 10. Let 1 ; 2 W Cp2 ! Z [ 1 be obgvs on Cp2 with 1 .g/  2 .g/ for all g 2 G. Prove that A.1 /  A.2 /. 11. Let  be a p-adic obgv on Cp2 . Prove that the sequence R ! H.i / ! A./ ! H.j / ! R is a short exact sequence of Larson orders. 12. Prove that H.i / is a local ring for 0  i < e 0 , and compute Spec H.i /. 13. Let F D HomR-alg .H.i /; / be the R-group scheme represented by H.i /. Compute F .R/. 14. Let K be a finite extension of Qp , and let H.i /, H.j / be Larson orders in KCp with i < j . (a) Show that there is an inclusion of R-algebras f W H.i / ! H.j /. (b) Show that H.j /=f .H.i /C /H.j / is not an R-Hopf order in K. 15. Prove Proposition 5.3.10.

Chapter 6

Formal Group Hopf Orders

In this chapter, we show how to construct Hopf orders using the theory of formal groups. Throughout this chapter, K D KP is a finite extension of Qp with ring of integers R D RO P endowed with the discrete valuation ord. We let  denote a uniformizing parameter for R, and put e D ord.p/, e 0 D e=.p  1/.

6.1 Formal Groups Definition 6.1.1. An n-dimensional formal group is an n-tuple of power series F .x; y/ D .F1 .x; y/; F2 .x; y/; : : : ; Fn .x; y// 2 RŒŒx; yn ; in the variables x D x1 ; x2 ; : : : ; xn , y D y1 ; y2 : : : ; yn , that satisfies, for i D 1; : : : ; n, (i) Fi .x; y/  xi C yi modulo degree 2, (ii) Fi .F .x; y/; z/ D Fi .x; F .y; z//, and (iii) F .x; 0/ D x, F .0; y/ D y. The formal group F is commutative if F .x; y/ D F .y; x/. Two formal groups F and G of dimension n are linearly isomorphic over R if there is a matrix ‚ in GLn .R/ such that G.x; y/ D ‚1 F .‚.x/; ‚.y//: An n-dimensional formal group F .x; y/ is a polynomial formal group if each power series Fi .x; y/ is a polynomial. We first consider one-dimensional polynomial formal groups. These are identified with a polynomial F .x; y/. We have the following classification. Proposition 6.1.1. Let F .x; y/ be a one-dimensional polynomial formal group. Then F is of the form F .x; y/ D x C y C axy for some a 2 R. R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 6, © Springer Science+Business Media, LLC 2011

115

116

6 Formal Group Hopf Orders

Proof. By Definition 6.1.1(i), F .x; y/ D x C y C g.x; y/, where g.x; y/ is a polynomial in RŒx; y in which all non-zero terms have total degree 2. Thus, we may write F .x; y/ D h.x; y/ C k.x; y/; where h.x; y/ has degree 1 in x and k.x; y/ has degree n for some n  1 in x. Now, by Definition 6.1.1(ii), h.F .x; y/; z/ C k.F .x; y/; z/ D h.x; F .y; z// C k.x; F .y; z//; and so the left-hand side above has degree n2 in x, while the right-hand side has degree n in x. These degrees must equal, and so n D 1 is the degree of x in F .x; y/. By a similar argument, the degree of y in F .x; y/ is also 1. Consequently, F is of the form claimed. t u Here are some examples of one-dimensional (polynomial) formal groups. Let F .x; y/ D x C y. Then, as one can easily show, F is a one-dimensional formal group, which is called the additive formal group, denoted by Ga . Another important example is the one-dimensional formal group defined as F .x; y/ D x C y C xy. Again, it is an easy exercise to show that F satisfies (i), (ii), and (iii) of Definition 6.1.1. This is the one-dimensional multiplicative formal group, which we denote by Gm . The following is an important consequence of Definition 6.1.1. Proposition 6.1.2. Let F .x; y/ be an n-dimensional formal group. Then there exists an n-tuple of power series in x, .x/ D .1 .x/; 2 .x/; : : : ; n .x//; for which F .x; .x// D 0 D F ..x/; x/: Proof. See [Fro68, Chapter 1, 3, Proposition 1].

t u

Formal groups are of interest because they allow us to put a group product on various sets of elements. As an example, we consider ./, the maximal ideal in R. We employ the formal group Gm .x; y/ D x C y C xy to define a multiplication on ./. For a; b 2 ./, put a  b D Gm .a; b/ D a C b C ab 2 : Then ./ is closed under . Moreover, zero serves as an identity element since Gm .a; 0/ D a D Gm .0; a/ by Definition 6.1.1(iii). But in order to obtain a group structure on ./, we need to define an inverse for a under . By Proposition 6.1.2, there exists a power series .x/ in RŒŒx for which Gm ..x/; x/ D 0 D Gm .x; .x//. Thus .x/ satisfies x C .x/ C x.x/ D 0;

6.1 Formal Groups

and so .x/ D

117

x . As a formal power series, 1Cx .x/ D x C x 2  x 3 C x 4     ;

and so the inverse of a is .a/ D a C a2  2  a3  3 C    ; which is a Œ p -Cauchy sequence converging to an element in ./. So ./ endowed with the group product defined by Gm .x; y/ is a group, which we denote by PGm . Proposition 6.1.3. PGm Š U1 .R/ D 1 C R, the group of principal units in R. Proof. Define a function  W PGm ! U1 .R/ by a 7! 1Ca. Then  is a bijection. Let a; b 2 ./. Then .a  b/ D .a C b C ab 2 / D ..a C b C ab// D 1 C .a C b C ab/ D .1 C a/.1 C b/ D .a/.b/; and so  is an isomorphism.

t u

For an integer j  1, there is a subgroup of U1 .R/ defined as Uj .R/ D f1 C a j W a 2 Rg: How does one obtain higher-dimensional formal groups? The easiest way is to construct an n-tuple consisting of copies of a given one-dimensional formal group. n For example, we have the n-dimensional formal group Gm defined as n .x; y/ D .Gm .x1 ; y1 /; Gm .x2 ; y2 /; : : : ; Gm .xn ; yn //; Gm

with Gm .xi ; yi / D xi C yi C xi yi for i D 1; : : : ; n. n We can modify Gm to create a collection of n-dimensional polynomial formal groups. Let ‚ be an n  n matrix with entries in R with det.‚/ ¤ 0. Consider the n-tuple of polynomials n .‚x; ‚y/ 2 KŒx; yn : F .x; y/ D ‚1 Gm

118

6 Formal Group Hopf Orders

Here x is the n1 column vector .x1 ; x2 ; : : : ; xn /T (T is the transpose); y is defined similarly. If the polynomials of F D F .x; y/ are in RŒx; y, then F is a formal group, which we call the n-dimensional generically split formal group determined by n ‚, denoted by F‚ . The formal group F‚ is linearly isomorphic to Gm over K. In what follows, we specialize to the case where ‚ is lower-triangular. To see n what kind of structure ‚1 Gm .‚x; ‚y/ has, we examine the cases n D 1; 2. For n D 1, ‚ is an element  6D 0 of R with ‚1 D  1 . Now,  1 Gm .x; y/ D  1 .x C y C  2 xy/ D x C y C xy; and this is precisely the form of all one-dimensional formal groups given by Proposition 6.1.1.    0 In dimension 2, we let ‚ D 1;1 denote a lower-triangular matrix with 2;1 2;2    0 2 .‚x; ‚y/ entries in R with det.‚/ 6D 0. Let ‚1 D 1;1 . Now, Gm 2;1 2;2 

 2 1;1 x1 C 1;1 y1 C 1;1 x1 y1 ; 2;1 x1 C 2;2 x2 C 2;1 y1 C 2;2 y2 C .2;1 x1 C 2;2 x2 /.2;1 y1 C 2;2 y2 /;

D

and so 2 ‚1 Gm .‚x; ‚y/ 

0  D 1;1 2;1 2;2



 2 1;1 .x1 C y1 / C 1;1 x1 y1 : 2;1 .x1 Cy1 /C2;2 .x2 Cy2 /C.2;1 x1 C2;2 x2 /.2;1 y1 C2;2 y2 /

The first component in this matrix product is x1 C y1 C 1;1 x1 y1 ; while the second component is 2;1 .1;1 .x1 C y1 / C 1;2 .x2 C y2 // C 2;1 .1;1 x1 C 1;2 x2 /.1;1 y1 C 1;2 y2 / C 2;2 .2;1 .x1 Cy1 /C2;2 .x2 Cy2 //C2;2 .2;1 x1 C2;2 x2 /.2;1 y1 C2;2 y2 / 2 x1 y1 D .2;1 1;1 C 2;2 2;1 /.x1 C y1 / C 2;2 2;2 .x2 C y2 / C 2;1 1;1

C 2;2 .2;1 x1 C 2;2 x2 /.2;1 y1 C 2;2 y2 / 2 2 C 2;2 2;1 /x1 y1 C 2;2 x2 y2 C 2;1 x1 y2 C 2;1 x2 y1 : D x2 C y2 C .2;1 1;1

6.1 Formal Groups

119

1 Now 2;2 D 2;2 and 2;1 D

2;1 , and so the coefficient on x1 y1 is 1;1 2;2

2 2;1 2;1 1;1 2;1 .2;1  1;1 / C D : 2;2 2;2 2;2 2 .‚x; ‚y/ Thus, ‚1 Gm 0

1 x1 C y1 C 1;1 x1 y1 A : (6.1) 2;1 .2;1  1;1 / D@ x2 C y2 C x1 y1 C 2;2 x2 y2 C 2;1 x1 y2 C 2;1 x2 y1 2;2

Of course, in order to define a two-dimensional formal group, we require that every coefficient in the components on the right-hand side of (6.1) be an element 2;1 .2;1  1;1 / , but it is not too of R. The only one that may not be in R is 2;2 difficult to find conditions such that this coefficient is in R. For example, as one 2 can verify, the conditions ord.1;1 / > ord.2;1 / and ord.2;1 /  ord.2;2 / guarantee 2;1 .2;1  1;1 / that 2 R. 2;2 We next consider the general case n  1. The goal is to find conditions on the n lower-triangular matrix ‚ 2 GLn .K/ such that ‚1 Gm .‚x; ‚y/ is defined over R. This is done in the paper [CU03, 1] of L. Childs and R. Underwood, and we review the result here. Proposition 6.1.4. Let ‚ D .i;j / 2 Mn .R/ be lower-triangular with det.‚/ 6D 0. p1 . Suppose the entries i;j of ‚ satisfy the Let q be a rational number 0 < q < 2p1 conditions (i) ord.i;i / > ord.i;j /  .1  q/ord.i;i / for all i > j with i;j 6D 0 and (ii) ord.r;r /  p  ord.rC1;rC1 / for all r. Then the formal group F‚ is defined over R. We will need the formal group F‚ constructed in Proposition 6.1.4 soon, but before that we discuss the types of maps that exist between formal groups. Definition 6.1.2. A homomorphism  W F ! G of n-dimensional polynomial formal groups is an n-tuple of polynomials .x/ D .1 .x/; 2 .x/; : : : ; n .x// for which i .F .x; y// D Gi ..x/; .y// for i D 1; : : : ; n. For example, let F .x; y/ D x C y C axy be a one-dimensional formal group over R, and let G D Gm .x; y/. Then  W F ! G, x 7! ax, is a homomorphism of formal groups since aF .x; y/ D ax C ay C a2 xy D Gm .ax; ay/:

120

6 Formal Group Hopf Orders

As another example, we describe a homomorphism of F into itself, an endomorphism of F . Define Œ1.x/ D x, Œ2.x/ D F .x; x/, and, for k > 2, Œk.x/ D F .Œk  1.x/; x/: Also, define Œ1.x/ D .x/, Œ2.x/ D F ..x/; .x//, and, for k > 2, Œk.x/ D F .Œk C 1.x/; .x//: Here .x/ is a power series for which F .x; .x// D 0 D F ..x/; x/. Proposition 6.1.5. Let F be an n-dimensional formal group. Then Œk W F ! F is an endomorphism for all non-zero integers k. t u

Proof. Exercise.

For example, the endomorphism Œ3 acts on Gm .x; y/ D x C y C xy as follows: Œ1.x/ D x D .1 C x/  1I Œ2.x/ D Gm .x; x/ D 2x C x 2 D .x C 1/2  1I Œ3.x/ D Gm .Œ2.x/; x/ D Gm .2x C x 2 ; x/ D 2x C x 2 C x C .2x C x 2 /x D 3x C 3x 2 C x 3 D .1 C x/3  1: For k > 0, one has the following proposition. Proposition 6.1.6. Let k > 0. Then the endomorphism Œk W Gm ! Gm is defined by .x/ D .1 C x/k  1. Proof. Exercise.

t u

There is a special type of formal group homomorphism that we will use in the next section to construct Hopf algebras. Definition 6.1.3. The homomorphism  W F ! G is an isogeny of formal groups if RŒŒx=..x// is a free R-module of finite rank. For example, Œp W Gm ! Gm is an isogeny since RŒŒx=..1 C x/p  1/ D RŒ1 C x=..1 C x/p  1/ D RŒy=.y p  1/ Š RCp : Clearly, RCp is free over R of rank p. In fact, it is an R-Hopf algebra. Here is an important generalization of the isogeny Œp W Gm ! Gm that we will need in the next section. Recall that Gm is the R-group scheme represented by the

6.2 Formal Group Hopf Orders

121

R-Hopf algebra RŒT; T 1 , with T indeterminate. We generalize this group scheme to define the group scheme Gnm of the form HomR-alg .RŒT1 ; : : : ; Tn ; T11 ; : : : ; Tn1 ; /; with Ti , i D 1; : : : ; n indeterminate. For n  1, we define a group scheme endomorphism of n W Gnm ! Gnm through the R-algebra homomorphism ‰n W RŒT1 ; : : : ; Tn ; T11 ; : : : ; Tn1  ! RŒT1 ; : : : ; Tn ; T11 ; : : : ; Tn1 ; given as ‰n .T1 / D T1 , ‰n .Ti / D Ti Ti1 1 , for i D 2; : : : ; n. Translating this to formal groups by setting xi D Ti  1, we obtain a homomorphism of formal groups p

p

n n ! Gm ;  n W Gm

defined as 0 1 0 1 p 1 .1 C x1 /p  1 T1  1 T1  1 p 1 p B 1 C B C C B T1 T2  1 C B .1 C x1 / .1 C x2 /  1 C B n .x/ D ‰n @ ::: A D B CDB C: :: :: @ A @ A : : Tn  1 1 p 1 p Tn1 Tn  1 .1 C xn1 / .1 C xn /  1 0

We have RŒŒx=.n .x// Š RCpn ; that is, n is an isogeny whose kernel is represented by the R-Hopf order RCpn in KCpn .

6.2 Formal Group Hopf Orders In this section, we show how to construct R-Hopf orders in KCpn from isogenies of formal groups. This collection of Hopf orders will include all of the Larson orders in KCpn . Our first step is to use an n-dimensional formal group F D .F1 ; : : : ; Fn / to induce a “formal” R-Hopf algebra structure on the ring of power series RŒŒx. One could imagine comultiplication on RŒŒx to be defined as .xi / D Fi .x ˝ 1; 1 ˝ x/; for i D 1; : : : ; n, but there is a problem here. It may be that Fi .x ˝ 1; 1 ˝ x/ 62 RŒŒx˝R RŒŒx for some i . For example, if Fi .x; y/ D x1 Cy1 Cx1 y1 Cx12 y12 C   , then Fi .x ˝ 1; 1 ˝ x/ D x1 ˝ 1 C 1 ˝ x1 C x1 ˝ x1 C x12 ˝ x12 C    ; which is not an element of RŒŒx ˝R RŒŒx. To remedy this, we need to replace RŒŒx ˝R RŒŒx with a larger ring, constructed as follows. Let I be the ideal of RŒŒx ˝ RŒŒx defined as I D .x1 ; x2 ; : : : ; xn / ˝ RŒŒx C RŒŒx ˝ .x1 ; x2 ; : : : ; xn /:

122

6 Formal Group Hopf Orders

Proposition 6.2.1. I is a prime ideal of RŒŒx ˝ RŒŒx. Proof. We show that .RŒŒx ˝R RŒŒx/=I Š R. Since .x1 ; x2 ; : : : ; xn / ˝ 1 2 I and 1 ˝ .x1 ; x2 ; : : : ; xn / 2 I , the quotient is isomorphic to R ˝R R Š R. u t Put T D RŒŒx ˝ RŒŒx. We endow T with the I -adic topology, a basis for which consists of all subsets of the form aCI  , where a 2 T and   0. A sequence fan g in T is I -Cauchy if for each I  there exists an integer N for which am  an 2 I  whenever m; n  N . We complete T with respect to the I -adic topology by adjoining the limits of all Cauchy sequences. The result is the completed tensor product and is denoted by O O RŒŒx˝RŒŒx. This is the larger ring that we require. For example, RŒŒx˝RŒŒx contains the I -convergent sum x1 ˝ 1 C 1 ˝ x1 C x1 ˝ x1 C x12 ˝ x12 C    ; which is not in T D RŒŒx ˝ RŒŒx. Proposition 6.2.2. Let F .x; y/ D .F1 .x; y/; : : : ; Fn .x; y// be an n-dimensional formal group. Then F induces a formal R-Hopf algebra structure on RŒŒx (formal O in the sense that the comultiplication will be a map W RŒŒx ! RŒŒx˝RŒŒx/. Proof. We define comultiplication, counit, and coinverse maps for RŒŒx and show that they satisfy the conditions for RŒŒx to be a formal R-Hopf algebra. We define O comultiplication W RŒŒx ! RŒŒx˝RŒŒx by the conditions .x/ D . .x1 /; : : : ; .xn //; O 1˝x/; O .xi / D Fi .x ˝1; O 1˝x// O O ˝1; O 1˝ .x//; O O D Fi .x ˝1 .I ˝ /.F i .x ˝1; O 1˝x// O O 1˝1 O ˝x/: O O /.Fi .x ˝1; D Fi . .x/˝1; . ˝I Now, O O 1˝x// O O .I ˝ /. .x i // D .I ˝ /.F i .x ˝1; O ˝1; O 1˝ .x// O D Fi .x ˝1 O ˝1; O 1˝F O .x ˝1; O 1˝x// O D Fi .x ˝1 O ˝1; O F .1˝x O ˝1; O 1˝1 O ˝x// O D Fi .x ˝1 O ˝1; O 1˝x O ˝1/; O O ˝x/ O 1˝1 by Def. 6.1.1(ii) D Fi .F .x ˝1 O 1˝x/ O ˝1; O 1˝1 O ˝x/ O D Fi .F .x ˝1; O 1˝1 O ˝x/ O D Fi . .x/˝1; O 1˝x// O O /.Fi .x ˝1; D . ˝I O /. .xi //; D . ˝I and so is coassociative.

6.2 Formal Group Hopf Orders

123

Next, we define the counit map W RŒŒx ! R by the conditions

.x/ D 0; O 1˝x// O O 1˝ .x//; O O D Fi .x ˝1; .I ˝ /.F i .x ˝1; O 1˝x// O O 1˝x/: O O /.Fi .x ˝1; D Fi . .x/˝1; . ˝I Then O / .xi / D m. ˝I O /.Fi .x ˝1; O 1˝x// O m. ˝I O 1˝x// O D m.Fi .0˝1; O i/ D m.1˝x

by Def. 6.1.1(iii)

D xi : O Likewise, m.I ˝ / .x i / D xi for all i . Thus satisfies the counit property. Finally, let .x/ D .1 .x/; : : : ; n .x// be the n-tuple of power series with F ..x/; x/ D 0 D F .x; .x//. Then the coinverse map  W RŒŒx ! RŒŒx is given as xi 7! .xi /. We leave it as an exercise to formulate the necessary conditions on  and to show that the coinverse property holds. t u RŒŒx together with the formal Hopf algebra structure induced by F is denoted by RŒŒxF . Proposition 6.2.3. Let  W F ! G, .x/ D .i .x/;    ; n .x//, be an isogeny of n-dimensional formal groups. Then  induces an R-Hopf algebra structure on the quotient RŒŒx=..x//. O denote the comultiplication map for the Proof. Let W RŒŒx ! RŒŒx˝RŒŒx O formal Hopf algebra RŒŒxF . One has an isomorphism W RŒŒx˝RŒŒx ! O O RŒŒx ˝1; 1˝x, and so induces an R-algebra map O 1˝x: O W RŒŒx ! RŒŒx ˝1; There exists an isomorphism O 1˝x O O O Š RŒŒx ˝1ŒŒ1 ˝x; RŒŒx ˝1; and consequently there exists a surjection O 1˝x O O O O ! .RŒŒx ˝1=..x ˝1///ŒŒ1 ˝x: RŒŒx ˝1; Since  is an isogeny, O O O O O O .RŒŒx ˝1=..x ˝1///ŒŒ1 ˝x Š RŒŒx ˝1=..x ˝1// ˝R RŒŒ1˝x: Thus there is a surjection O 1˝x O O O O RŒŒx ˝1; ! RŒŒx ˝1=..x ˝1// ˝R RŒŒ1˝x:

124

6 Formal Group Hopf Orders

Now the map induces a comultiplication map, O O O O ˝1// ˝R RŒŒ1˝x=..1 ˝x// RŒŒx=..x// ! RŒŒx ˝1=..x Š RŒŒx=..x// ˝R RŒŒx=..x//, which satisfies coassociativity since does. The counit and coinverse maps are constructed in a similar fashion.

t u

n To construct Hopf orders in KCpn , we modify the formal group Gm by the lower.p/ triangular matrices ‚ and ‚ in the manner of 6.1. This results in the formal groups F‚ and F‚.p/ . (‚.p/ is the matrix whose entries are the pth powers of those of ‚.) We then generalize the homomorphism n of 6.1 to give an isogeny F‚ ! F‚.p/ whose kernel is represented by a Hopf order in KCpn . The generalization of n is defined as

f .x/ D .‚.p/ /1 ‚n‚ .x/; where

n‚ .x/ D ‚1 n .‚x/:

We require certain integrality conditions on f .x/. Proposition 6.2.4. Let ‚ D .i;j / 2 Mn .R/ be lower-triangular with det.‚/ 6D 0. Suppose ord.i;i / > ord.i;j /  .1  q/ord.i;i / for all i > j with i;j 6D 0, where 0
p1 : 2p  1

Suppose also that ord.r;r / > 0 and ord.r;r /  d  ord.rC1;rC1 / for all r, with d  and 0

e >



q p C 1q 1  1q p

 p q  ord.1;1 /: 1C p1 d 1

Then f .x/ D .‚.p/ /1 ‚n‚ .x/ is in RŒŒx and satisfies f .x/  x .p/ mod RŒŒx: Proof. See the paper of L. Childs and R. Underwood [CU03, Theorem 2.0].

t u

With these preliminaries established, we can construct our formal group Hopf orders in KCpn .

6.2 Formal Group Hopf Orders

125

Proposition 6.2.5. Suppose ‚ is an n  n lower-triangular matrix with entries in R for which det.‚/ ¤ 0 and ord.r;r / > 0 for all r. Suppose for all r, and all s < r for which r;s ¤ 0, there exist numbers q and d such that ord.r;r / > ord.r;s /  .1  q/ord.r;r / and ord.r;r /  d ord.rC1;rC1 /, where p1 ; 2p  1    p1 d 1 ord.1;1 / < e0; p d 1Cq 0
and d 

p q : C 1q 1  1q p

Then ‚ gives rise to an R-Hopf order in KCpn . Proof. Put n F‚ .x; y/ D ‚1 Gm .‚x; ‚y/

and n .‚.p/ x; ‚.p/ y/: F‚.p/ .x; y/ D .‚.p/ /1 Gm

By Proposition 6.1.5, F‚ and F‚.p/ are defined over R, and, by Proposition 6.2.4, f .x/ D .‚.p/ /1 ‚n‚ .x/ is defined over R. Moreover, f W F‚ ! F‚.p/ is a homomorphism of generically split n-dimensional polynomial formal groups, and since f .x/  x .p/ mod RŒŒx, f is an isogeny. Applying Proposition 6.2.3, we conclude that the kernel of f is represented by an R-Hopf order in KCpn . t u The Hopf order given by the matrix ‚ of Proposition 6.2.5 is a formal group Hopf order and is denoted by H‚ . The structure of H‚ is determined as follows. Let ‚1 D .i;j /. Then H‚ D RŒz1 ; z2 ; : : : ; zn ; where z1 D 1;1 .g p z2 D 2;1 .g

n1

 1/

p n1

 1/ C 2;2 .g p

n2

 1/

n1

 1/ C n;2 .g p

n2

 1/ C    C n;n .g  1/;

:: : zn D n;1 .g p and hgi D Cpn . Note that the matrix ‚1 yields algebra H‚ .

n.n C 1/ parameters that describe the Hopf 2

Remark 6.2.1. Suppose ‚1 and ‚2 are lower-triangular matrices that satisfy the hypothesis of Proposition 6.2.5. Suppose ‚2 D ‚1 M , where M is in GLn .R/.

126

6 Formal Group Hopf Orders

1 Then .‚2 /1 D M 1 ‚1 is lower-triangular and corresponds to a 1 , where M 1 sequence of row operations on ‚1 . Hence, the associated Hopf orders H‚1 and H‚2 are equal.

Remark 6.2.2. Suppose ‚ is a diagonal matrix that satisfies the hypothesis of Proposition 6.2.5. Then, by Remark 6.2.1, we may assume ‚ D diag. i1 ; : : : ;  in / with ir1  pir for r D 2; : : : ; n since in this case we may take q D 0 and d D p. Now ‚1 D diag. i1 ; : : : ;  in /, and H‚ is a Larson order in KCpn whose exponents ir are given by a p-adic obgv on Cpn . It is fairly easy to satisfy the conditions of Proposition 6.2.5. Let n  3. Proposition 6.2.6. Let n  3, and let K be a finite extension of Qp such that .2p C 1/n C 1 : 2 Let ‚ be a lower-triangular n  n matrix with 8 .2pC1/nr if r D s < n2 1 .2pC1/ r;s D  if r D 2; s D 1 : 0 if r 6D s: eD

Put d D 2p C 1;

qD

1 : .2p C 1/n2

Then ‚ gives rise to an R-Hopf order in KCpn . Proof. We check that the conditions of Proposition 6.2.5 hold. We have ord.1;1 / D .2p C 1/n1 D

.2p C 1/n 2p C 1 <

.2p C 1/n C 1 2p C 1

.2p C 1/n C 1 2p C q    .2p C 1/n C 1 2 D 2p C q 2 <

2 e 2p C q    p1 2p D e0 p 2p C q    d 1 p1 e0: D p d 1Cq

D

6.2 Formal Group Hopf Orders

127

Moreover, q < 1, and so q < 1 

1q . p

Thus

p p q : C1 > C 1q 1q 1  1q p p . Thus 1q p q p : C1> C d D 2p C 1 > 1q 1q 1  1q p

Also, since q < 1=2, 2p >

In addition, ord.2;1 / D .2p C 1/n2  1 .2p C 1/n2  1 .2p C 1/n2 .2p C 1/n2   1 D 1 .2p C 1/n2 .2p C 1/n2 D

D .1  q/ord.2;2 /; and ord.r;r / D .2p C 1/ord.rC1;rC1 /: Thus ‚ satisfies the conditions of Proposition 6.2.5, and consequently there exists an R-Hopf order in KCpn of the form H‚ . t u To illustrate Proposition 6.2.6, take p D 5, n D 4. Let K D Q5 .˛/, where ˛ 7321 D 5. We have 5R D  e with e D 7321. Then ord.5/ D 7321. Let ‚ be the matrix 0 1331 1  0 0 0 B  120  121 0 0C C: ‚DB 11 @ 0 0  0A 0 0 0  Then

0

‚1

 1331 B 1332 DB @ 0 0

0  121 0 0

0 0  11 0

1 0 0 C C; 0 A  1

and thus H‚ is an R-Hopf order in KC625 of the form  g 125  1 g125  1 g 25  1 g5  1 g  1 : H‚ D R ;  ; ;  1331  1332  121  11  

128

6 Formal Group Hopf Orders

6.3 Chapter Exercises Exercises for 6.1 1. Let F .x; y/ be a formal group. Show that conditions (i) and (ii) of Definition 6.1.1 together with the condition F .x; 0/ D x imply that F .0; y/ D y. 2. Show that F .x; y/ D x C y defines a one-dimensional formal group. Show that F .x; y/ D x C y C xy defines a one-dimensional formal group. 3. Let F .x; y/ D x C y C x 2 C xy C y 2 C x 3 C x 2 y C xy 2 C y 3 C    . Does F .x; y/ determine a one-dimensional formal group? 4. Let F .x; y/ be a one-dimensional polynomial formal group. Find a power series .x/ for which F .x; .x// D 0. 5. Consider the polynomial formal group x1 C y1 C 1;1 x1 y1 F .x; y/ D x2 C y2 C 2;1 .2;12;21;1 / x1 y1 C 2;2 x2 y2 C 2;1 x1 y2 C 2;1 x2 y1

!

constructed in 6.1. Find a 2-tuple of power series .x/ D .1 .x/; 2 .x// for which F .x; .x// D 0. 6. Let ‚ be a matrix that satisfies the hypothesis of Proposition 6.1.4. Show that, for any prime p, p  ord.i;j /  ord.i;i / for all i > j . 7. Prove Proposition 6.1.5. 8. Prove Proposition 6.1.6. Exercises for 6.2 9. Finish the proof of Proposition 6.2.2 by formulating the necessary conditions on  and showing that the coinverse property holds.

Chapter 7

Hopf Orders in KCp

Let p be a rational prime, let K be a finite extension of Q, and let G be a finite Abelian group. In Chapter 5, we constructed a collection of Hopf orders in KG using p-adic order-bounded group valuations on G. These were called Larson orders. We specialized to the case G D Cpn , completed R at the prime P above p, and considered Larson orders over the complete local ring RO P . In Chapter 6, we constructed a collection of formal group Hopf orders in KP Cpn and found that the Larson orders in KP Cpn formed a proper subcollection of the formal group Hopf orders. In this chapter, we assume that K is a finite extension of Qp containing p , with ring of integers R, uniformizing parameter , and discrete valuation ord. Note that .p/ D P p1 , where P D .1  p /, and so e 0 D ord.p/=.p  1/ is an integer. We give a complete classification of Hopf orders in KG, where G is the cyclic group of order p.

7.1 Classification of Hopf Orders in KCp Let g be a generator for Cp . Then a Larson order in KCp can be written  H.i / D R

 g1 ; i

where i is an integer 0  i  e 0 . In this section, we show that every Hopf order in KCp is of the form H.i / for some integer i , 0  i  e 0 . We begin with the classification of rank p Hopf algebras given by J. Tate and F. Oort [TO70], following closely the notes of L. Childs [Ch00]. Let H be an R-Hopf order in KCp . For each n 2 F p D f1; 2; 3; : : : ; p  1g, there is a Hopf algebra endomorphism Œn W H ! H defined as Œn.h/ D m.n1/ ..n1/ .h// R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 7, © Springer Science+Business Media, LLC 2011

129

130

7 Hopf Orders in KCp

for h 2 H (See 3.4). The collection of endomorphisms E D fŒn W n 2 F pg is a group under the operation ŒnŒn0  D Œnn0 , with nn0 taken modulo p. Clearly  E Š F p , and henceforth we identify Fp with this collection of endomorphisms. We  have the group ring Zp Fp . By Hensel’s Lemma, Zp contains the .p  1/st roots of unity. For each Œn 2 F p, there exists a root of X p1 1 in Zp that is congruent to n modulo p. This defines the Teichmuller ¨ character, which is the unique multiplicative group homomorphism  p1  W F ! Z  1, which satisfies .n/  n p p , where .n/ is the root of X .mod p/. For j , 1  j  p  1, put j D

1 X j 1  .n/ Œn: p1  n2Fp

Then the j form a set of pairwise mutually orthogonal minimal idempotents in Zp F p such that Zp F p D Zp 1 ˚ Zp 2 ˚    ˚ Zp p1 : Let H C denote the augmentation ideal of H . From the short exact sequence

0 ! H C ! H ! R ! 0; C C one obtains H D R ˚ H C . Moreover, Zp F p .H / D H , and so

H C D 1 .H C / ˚ 2 .H C / ˚    ˚ p1 .H C /:

(7.1)

Lemma 7.1.1. For i; j , 1  i; j  p  1, we have i .H C /j .H C /  i Cj .H C /, where the subscript i C j is taken modulo p  1. Proof. Let h 2 H C . Then Œni .h/ D i .n/.h/ for all 1  i  p  1, n 2 F p . Thus n o i .H C / D h 2 H C W Œn.h/ D i .n/h; 8n 2 F p : Let hi 2 i .H C /, hj 2 j .H C /. Then, for all Œn 2 F p, Œn.hi hj / D Œn.hi /Œn.hj / D i .n/hj .n/hi hj D i Cj .n/hi hj ; so that hi hj 2 i Cj .H C /.

t u

7.1 Classification of Hopf Orders in KCp

131

With these preliminaries in mind, we give the Tate/Oort classification of Hopf orders in KCp . We begin with a characterization of the Hopf order H D RCp in KCp . Lemma 7.1.2. Let

xD

X

.n/1 .g n  1/:

n2F p

Then RCp D RŒx with x p D wx for some w 2 R. Proof. By (7.1), there is a decomposition RCp D R ˚ 1 .RCpC / ˚ 2 .RCpC / ˚    ˚ p1 .RCpC /:

(7.2)

Let k D R=R be the residue class field of R, and set kCpC D k˝R RCpC . By direct calculation, 1 .kCpC / D k ˝R 1 .RCpC / D kx, and from this and Lemma 7.1.1 one deduces that i .kCpC / D kx i for 1  i  p  1. It follows that i .RCpC / D Rx i for 1  i  p  1, so that RCp D RŒx by (7.2). By Lemma 7.1.1 x p 2 1 .RCpC /, and thus x p D wx for some w 2 R. t u We give a similar characterization for the dual RCpD , which by Proposition 4.4.8 p1

is an R-Hopf order in KCp . Let fei gi D0 denote the set of minimal idempotents in KCp . Then RCpD D Re0 ˚ Re1 ˚    ˚ Rep1 . P Lemma 7.1.3. Let y D n2Fp .n/en . Then i ..RCpD /C / D Ry i , for i D 1; : : : ; p p  1, and RCD p D RŒy with y D y.

t u

Proof. For a proof, see [Ch00, (16.10)]. Now let H be an arbitrary Hopf order in KCp . Lemma 7.1.4. H D RŒz with zp D bz for some b 2 R, z 2 H . Proof. We have the decomposition H D R ˚ 1 .H C / ˚ 2 .H C / ˚    ˚ p1 .H C /;

with 1 .H C / D Rz for some z 2 H , by [Ch00, (16.12)]. Let k be the residue class field of R, and let k be an algebraic closure of k. By [Wat79, 6.8], kH D k ˝R H is either separable as a k-algebra or a local ring. In the separable case, one has kH D k ˝k kH Š k ˚ k ˚    ˚ k „ ƒ‚ … p

so that kH D kCpD . It follows that H D RCpD , and so H D RŒz with zp D bz; b in R, by Lemma 7.1.3. If kH is local with separable dual, then kH D kCp , and so H D RŒz, with zp D bz; b 2 R by Lemma 7.1.2.

132

7 Hopf Orders in KCp

In the case where kH and kH D are local, we have kH Š kŒt with t p D 0 by [Wat79, 14.4]. As argued in [Ch00, (16.13)], one deduces that i .kH C / Š i .kŒtC / D kt i for 1  i  p  1, and thus i .H C / D Rzi for 1  i  p  1, so that H D RŒz. Now, by Lemma 7.1.1, zp 2 1 .H C / D Rz, so that zp D bz for some b 2 R. t u We can now give the Tate/Oort classification of Hopf orders in KCp . Proposition 7.1.1. Let H be a Hopf order in KCp . Then H D RŒz, where zp D bz with 1 X .n/1 .g n  1/ zD c  n2Fp

for some b; c 2 R. Proof. By Lemma 7.1.4, H D RŒz for some z 2 H , with zp D bz, b 2 R, and, by Lemma 7.1.2, RCp D RŒx with x p D wx for w 2 R with X .n/1 .g n  1/ : xD n2F p

Let L be an extension of K whose ring of integers S contains the .p  1/st roots of w and b. Let c 2 S be such that c p1 D w=b. Then S Cp D S Œx embeds into SH through the relation x D cz. It follows that H D RŒz with zp D bz and 1 X zD .n/1 .g n  1/: c t u n2F p Using the Tate/Oort classification, we can show that every Hopf order in KCp is a Larson order. Let H be an arbitraryP R-Hopf order in KCp . By Proposition 7.1.1, H D RŒz and zp D bz, with z D  1c n2Fp .n/1 .g n  1/. We have zD

1 X .n/1 .g n  1/ c  n2Fp

D

.g  1/ X .n/1 .1 C g C g 2 C    C g n1 / c  n2Fp

.g  1/ D v c with vD

X

  .n/1 1 C g C g2 C    C gn1 :

n2F p

Observe that v 2 Zp Cp  RCp .

7.1 Classification of Hopf Orders in KCp

133

Lemma 7.1.5. v is a unit in Zp Cp . Proof. Let X be indeterminate. By Lemma 7.1.3, Zp CpD D Zp ŒX =.X p  X / Š Zp ˚ Zp Œp : There is an embedding Zp Cp ! Zp ˚ Zp Œp  under this embedding. We have defined by g 7! .1; p /. Let P vO denote the image of v P vO D .˛; ˇ/, where ˛ D n2Fp .n/1 n and ˇ D n2Fp .n/1 .1 C p C p2 C    C pn1 /. If ˛ is not a unit of Zp , then X

.n/1 n  0 .mod p/:

n2F p

Since .n/  n .mod p/, this says that p  1 .mod p/, which is impossible. Thus ˛ is a unit of Zp . Moreover, if ˇ is not a unit of Zp Œp , then there exists an integer k  0 such that 0 @

X

1k .n/1 .1 C p C p2 C    C pn1 /A  0 .mod .1  p //:

n2F p

Since 1  pi  0 .mod .1  p // for all i , 1  i  n  1, 0 @

X

1k .n/1 nA  0 .mod .1  p //;

n2F p

and thus there is an integer s  0 with 0 @

X

1st .n/1 nA  0 .mod p/;

n2F p

which again leads to a contradiction. Thus ˇ is a unit in Zp Œp . It follows that vO is a unit in Zp CpD . Let y 2 Zp CpD be such that vO y D 1. Since y is integral over Zp , there exists a monic polynomial f .X / D X l C a1 X l1 C    C al1 X C al with y l C a1 y l1 C    C al1 y C al D 0. Multiplying this equation by vO l1 yields y C a1 C a2 vO C    C al vO l1 D 0:

134

7 Hopf Orders in KCp

Hence, y D a1  a2 vO      al vO l1 : Since v 2 Zp Cp , r D a1  a2 v      al vl1 2 Zp Cp with vr D 1. Thus v is a unit of Zp Cp  RCp . t u We can now show that every Hopf order in KCp is a Larson order. Proposition 7.1.2. Let p be a prime number, and let K be a finite extension of Qp with ring of integers R and uniformizing parameter . Suppose p 2 K. Let e 0 D ord.p/=.p  1/. Let Cp denote the cyclic group of order p, generated by g, and let H be an R-Hopf order in KCp . Then   g1 ; H DR i where i is an integer 0  i  e 0 .

 v.g  1/ , where v is defined as above and Proof. By Proposition 7.1.1, H D R c c is some element of R. By Lemma 7.1.5, v is a unit of RCp . Now,  by Proposition  g1 4.4.3, RCp  H , and so v is a unit of H . It follows that H D R . c Let c D u i for some unit u 2 R,integer i  0, where  is a uniformizing g1 . It remains to show that 0  i  e 0 . By parameter for R. Then H D R i p1 Proposition 4.4.9, H embeds into RCpD through g 7! .1; p ; p2 ; : : : ; p /, and 0 consequently ord.p  1/ D e  i  0. t u 

Proposition 7.1.2 can be applied in the following way. Let hgi D Cpn , so that n1 i D Cp . Let H be an R-Hopf order in KCpn , n  1. Then Khgp i \ H is hg n1 an R-Hopf order in KCp . Consequently, Khg p i \ H is the Larson order # " n1 gp  1 H.i / D R i pn1

for some integer 0  i  e 0 . This fact is used in the following proposition. Proposition 7.1.3. Let n  1 be an integer, and suppose that K contains a primitive pn th root of unity pn . Let H be an R-Hopf order in KCpn , let H.i / denote the Larson order as above, and let H denote the image of H under the canonical  n1 surjection g p 7! 1. Let H denote the linear dual of H . Let CO pn D h”i denote the character group of Cpn . Set J DH





 ”u  1 ; i0

with u 2 Kh” p i, i 0 D e 0  i . If hJ; H i  R, then J D H  .

7.1 Classification of Hopf Orders in KCp

135

Proof. From hJ; H i  R, one has J  H  . We show that J DH  by showing that their discriminants are equal. Let ˛ D ”u1 0 . We have i 0

”p D

.1 C ˛ i /p 2 Kh” p i; up

and so, since e D .p  1/e 0  .p  1/i 0 , ! p1 X ” p up  1 p 0 ˛p C 2 H  \ Kh” p i: ˛ r  .rp/i D pi 0 r  rD1 

Since H D H  \ Kh” p i, ˛ satisfies a monic degree p polynomial with   p n1 coefficients in H . Consequently, if fa g D1 is an R-basis for H , then fa ˛ k g n1 with D 1; 2 : : : ; p , k D 0; 1; : : : ; p  1, is an R-basis for J . There is a short exact sequence of Hopf orders, R ! H.i / ! H ! H ! R; that dualizes to yield the short exact sequence of duals 

R ! H ! H  ! H.i / ! R: By Proposition 4.4.11, Z



H.i /

Z H.i /

 H.i /

H.i /

D pR;

R and so, by Proposition 5.3.3, H.i / H.i / D  .p1/i R. Therefore, by Proposition 4.4.12, Z  Z  D H   .p1/i R: H   H

Thus,

Z H 

H

pn H

Z D H 

pn H



p

n .p1/i

R:

Now, by Proposition 4.4.13, 

disc.H  / D disc.H /p  p

n .p1/i

R:

(7.3)

  Let J0 D H Œu”  1 D H Œ”. Then J0 is an R-Hopf order in K CO pn , and we can compute the discriminant of J0 using the exact sequence 

R ! H ! J0 ! H.0/ ! R:

136

7 Hopf Orders in KCp

We have 

disc.J0 / D disc.H /p  p

n .p1/e 0

R:

(7.4)

Let 0

0

0

0

0

0

M D diag.1; 1; : : : ; 1;  i ;  i ; : : : ;  i ; : : : ;  .p1/i ;  .p1/i ; : : : ;  .p1/i /: ƒ‚ … „ ƒ‚ … „ ƒ‚ … „ p n1

p n1

pn1

Then M multiplies the basis fa ˛ k g of J to give the basis fa .u”/k g of J0 . Consequently, disc.J0 / D det.M /2 disc.J / D p

n .p1/i 0

disc.J /:

Now disc.J / D disc.J0 / p

n .p1/i 0

R

 p

p n .p1/e0 p n .p1/i 0

 p

p n .p1/i

D disc.H /  D disc.H /  

D disc.H /

R

by (7.4)

R

by (7.3):

Since J  H  and their discriminants are equal, J D H  .

t u

By Proposition 4.4.8, the linear dual of the R-Hopf order H.i / in KCp is an R-Hopf order H.i / in K CO p , h”i D CO p . We compute H.i / . Proposition 7.1.4. Let H.i / be an R-Hopf order in KCp . Then H.i / D H.i 0 / D

 0 0 R ”1 0 , where i D e  i . i

 Proof. Let J DR ”1 be the Larson order given by the parameter i 0 . For 0 i  a D 1; : : : ; p  1,   .p  1/a ”1 g1 a ; ; D 0 i i  i 0 Cai which is in R since ord.p  1/ D e 0 . Thus J  H.i / . Now, by Proposition 7.1.3 with n D 1, u D 1, one has H.i 0 / D H.i / . t u Proposition 7.1.3 can also be used to compute the linear dual of a Larson order in KCp2 . Proposition 7.1.5. Assume that p2 2 K, let CO p2 D h”i denote the character group of Cp2 , and let H.i; j / be a Larson order in KCp2 . Then

7.1 Classification of Hopf Orders in KCp 

0

H.i; j / D A.j ; i

0

137

; p1 2 /



 ” p  1 ”u  1 DR ; ; j 0 i0

Pp1 where m D aD0 pma ” a .

 Proof. Note that H.i / D Khg p i \ H.i; j / and H.i; j / D R g1 , where H.i; j / j denotes the image of H.i; j / under the canonical surjection gp 7! 1. By Proposition

p   7.1.4, H.i; j / D R ” j1 . Put 0 where i 0 D e 0  i , j 0 D e 0  j , and u D



Pp1

”p  1 J DR j 0

m mD0 p2 m ,



 ”u  1 : i0

Then, as one can easily compute, for 0  s; t  p  1, ˝

˛ 0 ” p  1; .g p  1/s .g  1/t 2  j Csi Ctj R;

and, for 0  s; t  p  1, ˛ ˝ 0 ”u  1; .gp  1/s .g  1/t 2  i Csi Ctj R: Thus J  H.i; j / , and so J D H.i; j / by Proposition 7.1.3. By Proposition 4.4.9, the R-Hopf order H.i / can be embedded into by the mapping g 7!

p1 X mD0

pm em ;

em D

Lp1 mD0

t u Rem

p1 1 X ma a  g : p aD0 p

Thus every element h 2 H.i / can be written as an R-linear combination of the Pp1 idempotents em . We ask: When does a linear combination mD0 am em , am 2 R determine an element h 2 H.i /? C. Greither [Gr92] has provided the following answer. Pp1 Lemma 7.1.6. Let a D mD0 am em , am 2 R. Then a 2 H.i / if and only if all of the following conditions hold: ord.a0 /  0, ord.a1  a0 /  i 0 , ord.a2  2a1 C a0 /  2i 0 , ord.a3  3a2 C 3a1  a0 /  3i 0 , :: : P p1 p1 m 0 ord a .1/ p1m  .p  1/i . mD0 m

138

7 Hopf Orders in KCp

Proof. By Proposition 7.1.4, H.i / D H.i 0 / D R only if

”1  i

0

. Thus a 2 H.i / if and

hH.i 0 /; ai  R: But this is equivalent to

*

”1 i0

+ m X p1 ; am em mD0

for m D 0; : : : ; p  1. Expanding the powers of h” m ; en i D ımn yields a 2 H.i / if and only if ord

k X mD0

”1 0 i

and using the identity

! ! k m .1/ akm  ki 0 m

for k D 0; : : : ; p  1.

t u

The next lemma, also due to Greither, shows that we can extend the “partial” sum †lmD0 am em , l < p  1 to obtain an element of H.i /. Lemma 7.1.7. Assume that l < p  1. Let a D †lmD0 am em , am 2 R. Suppose that a satisfies ! ! k X k ord .1/m akm  ki 0 m mD0 lC1 for k D 0; : : : ; l. Then there exists an element alC1 2 R for which a0 D †mD0 am em satisfies ! ! k X k m .1/ akm  ki 0 ord m mD0

for k D 0; : : : ; l C 1. PlC1 lC1 .1/m alC1m . Then b D u r for some unit u 2 R and Proof. Put b D mD1 m integer r  0. Let alC1 D 1 C  s  u r , where s  minfr; .l C 1/i 0 g. Then ! ! lC1 X l C1 m .1/ alC1m  .l C 1/i 0 ; ord .alC1 C b/ D ord m mD0 which proves the proposition.

t u

By using Lemma 7.1.7 repeatedly, one can extend the sum †lmD0 am em to obtain an element of H.i /.

7.2 Chapter Exercises

139

7.2 Chapter Exercises Exercises for 7.1 1. Prove that Fp Š Zp =pZp . n 2. Let O W Zp ! Zp be the map defined as .a/ O D limn!1 ap ; 8a 2 Zp . (a) Show that O is a ring homomorphism. (b) Prove that .a/ O D .a .mod p//; 8a 2 Zp ; that is, show that the image of O is the same as the image of the Teichm¨uller character. R 3. Let H.i / be a Hopf order in KCp . Compute the ideal of integrals H.i / . 4. Let g p 7!1 K ! Khg p i ! KCp2 ! Khgi ! K denote the short exact sequence of K-Hopf algebras, and suppose that H is an R-Hopf order in KCp2 . Show that there exists a short exact sequence of R-Hopf orders R ! H.i / ! H ! H.j / ! R; where H.i / and H.j / are Larson orders in KCp with i  j .

Chapter 8

Hopf Orders in KCp2

In this chapter, we assume that K is a finite extension of Qp , containing p2 , endowed with the discrete valuation ord. Set e D ord.p/, e 0 D e=.p  1/. Let g be a generator for Cp2 .

8.1 The Valuation Condition The K-Hopf algebra KCp2 induces the short exact sequence of K-Hopf algebras 

i

s



K ! KCp ! KCp2 ! KCp ! K; where i W KCp ! KCp2 is the Hopf inclusion and s W KCp2 ! KCp is the Hopf surjection, given as g p 7! 1. Let H denote an R-Hopf order in KCp2 . Since H 0 D H \ KCp is an R-Hopf order in KCp and H 00 D s.H / is an R-Hopf order in KCp , one has the short exact sequence of Hopf orders R ! H 0 ! H ! H 00 ! R:

(8.1)

By Proposition 7.1.2, H 0 and H 00 are Larson orders in KCp of the form 

 gp  1 ; H D H.i / D R i 0



 g1 H D H.j / D R ; g D s.g/; j 00

where i; j are integers satisfying 0  i; j  e 0 . Thus (8.1) can be written as R ! H.i / ! H ! H.j / ! R:

(8.2)

By Proposition 4.3.3, H has a generating integral ƒ, which we now compute.

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 8, © Springer Science+Business Media, LLC 2011

141

8 Hopf Orders in KCp 2

142

Proposition 8.1.1. The ideal of integrals ƒD where e0 D

1 p2

Pp2 1 mD0

R H

is of the form Rƒ, where

p2  .p1/.i Cj /

e0 ;

gm .

R R  Proof. By Proposition 4.4.5, H D H H e0 , and so, by Proposition 4.4.12, Z  Z  Z D H.i / H.j / e0 : H

Now, by Proposition 5.3.3, H.i / R Thus H D Rƒ with

H.i /

R



H.i /

ƒD

H.j /

D

p

R, and H.j /  .p1/i

R



H.j /

p2

e0 :  .p1/.i Cj /

D

p  .p1/j

R.

t u

By Proposition 7.1.4, 

 ”p  1 ; H.j / D R j 0 0

0

H.i / D R



 ”1 ; i0

and the sequence (8.2) can be dualized (see (4.13)) to form the short exact sequence s

i

R ! H.j 0 / ! H  ! H.i 0 / ! R: Definition 8.1.1. Let H be an R-Hopf order in KCp2 that induces the short exact sequence (8.2). Then H satisfies the valuation condition for n D 2 if either pj  i or pi 0  j 0 . In the paper [Lar88], R. Larson proved that every Hopf order in KC4 satisfies the valuation condition. In [Un94], R. Underwood showed that the valuation condition holds for Hopf orders in KCp2 , p  2. Underwood’s result also follows from results of N. Byott; see [By93b, 8, Theorem 5]. In this section, we prove that the valuation condition holds for Hopf orders in KCp2 , following closely the proof in [Un94]. We first prove that the valuation condition holds for H that induces short exact sequences (8.2) with i  e=p. The key is to obtain an R-basis for H  in a special form. Lemma 8.1.1. Let H be an R-Hopf order in KCp2 inducing the short exact p sequence (8.2) with i  e=p. Let h D ” 1 , k D ”1 0 . Put e0 i  X D k; k 2 ; : : : ; k p1 ; hk; hk 2 ; : : : ; hk p1 ; : : : ;

 hp1 k; hp1 k 2 ; : : : ; hp1 k p1 ; 1; h; : : : ; hp1 :

8.1 The Valuation Condition

143 p2

Then there exists an R-basis f˛m gmD1 for H  given by the matrix product XM 0 D .˛1 ; ˛2 ; : : : ; ˛p2 /; where M 0 is the p 2  p 2 matrix 0 r1;1 B 0 B M0 D B : @ :: 0

r1;2 r2;2 :: :

 

 

r1;p2 r2;p2 :: :

0





rp2 ;p2

1 C C C; A

which satisfies the following conditions: (i) either rl;m D 0 or ord.rl;m / > ord.rl;mC1 / for all 1  l; m  p 2  1; (ii) ord.rl;m /  aj C bi 0  .p  1/i 0 for 1  m  p 2  p, where l D a.p  1/ C b for 0  a  p  1, 1  b  p  1. Proof. Since H.i / injects into H , and RCp2  H , the Larson order H.i; 0/ is contained in H . Thus   p ”  1 ”u  1   ; H  H.i; 0/ D R ;  e0 i0 where u D †mD0 pm 2 m by Proposition 7.1.5. But, since i  e=p, Lemma 7.1.6 implies that p1 u1 1 X  m D  1 m 2 H.e 0 /:  2 0 0 i  i mD0 p p1

Thus



  p  ” p  1 ”u  1 ” 1 ”1 ; ; D R ; 0 0 0 0 e i e i h p i ; ”1 . One has that X is an R-basis for H.e 0 ; i 0 /. and so H   H.e 0 ; i 0 / D R ” 1 0 e0 i  R

p2

Therefore, an R-basis f˛m gmD1 for H  is given by the matrix product XM D .˛1 ; ˛2 ; : : : ; ˛p2 /; where M is a p 2  p 2 matrix with entries in R. Performing elementary column operations on M , one sees that it is column-equivalent to the matrix 0

r1;1 B 0 B M0 D B : @ :: 0

r1;2 r2;2 :: :

 

 

r1;p2 r2;p2 :: :

0





rp2 ;p2

1 C C C; A

8 Hopf Orders in KCp 2

144

where either rl;m D 0 or ord.rl;m / > ord.rl;mC1 / for all 1  l; m  p 2  1. Thus p2 f˛m gmD1 defined by XM 0 D .˛1 ; ˛2 ; : : : ; ˛p2 / is an R-basis for H  . Note that M 0 satisfies condition (i) of the lemma. We now show that M 0 satisfies condition (ii). We have p 2 p

2

X

˛m D

p X

rl;m h k C a b

r;m hp

2 Cp1

Dp 2 pC1

lDa.p1/Cb

for m D 1; : : : ; p 2 , 0  a  p  1, 1  b  p  1. Since M 0 is upper-triangular, p2 rl;m D 0 for l D m C 1; : : : ; p 2 . By Proposition 8.1.1, ƒ D  .p1/.i Cj / e0 is a generating integral for H . Thus, by Proposition 4.3.4, ˛m ƒ 2 H for m D 1; : : : ; p 2 . Let dm D ˛m  ƒ. Then 0

p 2 p

X

dm D

a.p1/CbD1 2

C

1 ! ! ! a b X 1 1 X b a   @ A .1/ e.a/p .1/ eb  ae0 D0   bi 0 D0 

p2 rl;m  .p1/.iCj / 0

p X

p2 r;m

Dc

 .p1/.iCj /

@

1  .c/e0

1 ! c X c .1/ e.c/p A ;  D0

where c D p 2  p C 1. We have H.j 0 /  H  , so that H.j 0 ; 0/  H  and H  H.j 0 ; 0/ . By p2 0  Proposition 8.1.1, % D  .p1/.e 0 Cj / e0 is a generating integral for H.j ; 0/ , and so, 0  by Proposition 4.3.4, a basis for H.j ; 0/ can be written as (

where

and

”p  1 j 0

a 

”1 0

b

) % D



 ; A B .p1/.e 0 Cj / a b p2



(8.3)

! a 1 X a Aa D aj 0 .1/ e.a/p  D0  ! b X b Bb D .1/ eb  D0

for 0  a; b  p  1. Now, since dm 2 H  H.j 0 ; 0/ , dm can be written as an integral linear combination of the basis (8.3) for H.j 0 ; 0/ . Let m be such that 1  m  p 2  p. Each term of dm can be written as

8.1 The Valuation Condition

p 2 rl;m  .p1/.i Cj /

! ! ! ! a b X 1 X a 1 b .1/ e.a/p .1/ eb  ae0 D0   bi 0 D0  0

D

rl;m  aj  e .p1/i   bi 0  ae0 0

and so

rl;m  aj  e 0 0  .p1/i  bi  ae

145



 A B ; .p1/.e0 Cj / a b p2



2 R or 0

0

0

ord.rl;m  aj  e /  ord. .p1/i  bi  ae /; ord.rl;m / C aj 0 C e  .p  1/i C bi 0 C ae 0 ; ord.rl;m /  aj C bi 0  .p  1/i 0 :

t u

We can now show that the valuation condition holds in the case i  e=p. Lemma 8.1.2. Let H be an R-Hopf order in KCp2 that induces the short exact sequence (8.2). If i  e=p, then pi 0  j 0 . p2

Proof. Let fam gmD1 denote the R-basis for H  as constructed in Lemma 8.1.1. ” p 7!1

Let ˛m denote the image of ˛m under the R-module surjection H  ! H.i 0 /. Since f˛m g is an R-basis for H.i 0 /, we can assume that r1;m0 D 1 for some m0 , 1  m0  p 2 . Since H  D S ˚ H.j 0 / for some R-module S , Lemma 8.1.1(i) implies that 0 m  p 2  p, and so, since M 0 is upper-triangular, we have rl;m0 D 0 for all l  p 2  p C 1. Thus the basis element ˛m0 2 H  has the form p 2 p

p 2 p

X

˛m 0 D

X

rl;m0 h k D k C a b

lDa.p1/CbD1

rl;m0 ha k b :

lDa.p1/CbD2

Thus, H  contains the pth power p 2 p p ˛m0

Dk C p

X

p rl;m0 hap k bp

C

X P

lDa.p1/CbD2

p 2 p

Y

puP

.rl;m0 ha k b /nl ;

lDa.p1/CbD1

where the second summation is over all partitions P of p that are in the form p D p2 p †lD1 nl , nl  0, and uP is a unit of R dependent on the partition P. We claim that X P

p2 p

puP

Y

lDa.p1/CbD1



rl;m0 ha k b

nl

2 H :

8 Hopf Orders in KCp 2

146

By Lemma 8.1.1(ii), ord.rl;m0 /  aj C bi 0  .p  1/i 0 , and so there exist elements sl 2 R for which 0 sl  aj  bi 0 rl;m D .p1/i 0 :  Thus X

p 2 p

Y

puP

P

X

.rl;m0 ha k b /nl D

p 2 p

P

a.p1/CbD1

P

P

p 2 p

Y

puP  p.p1/i 0

X

D

0

sl  aj  bi a b h k  .p1/i 0

a.p1/CbD1

X

D

Y

puP

0

n

sl l . aj ha  bi k b /nl

a.p1/CbD1



p 2 p

Y

puP  p.p1/i

!n l

slnl

0

a.p1/CbD1

” p 1 0 j

anl 

”1 0

bnl ;

which is in H  since i  e=p implies e 0 =p  i 0 , which in turn yields ord.p/ D e  p.p  1/i 0 . We conclude that p 2 p

X

k C p

rl;m0 hap k bp 2 H  ; p

lDa.p1/CbD2

and thus 

”1 i0

p



p2 p

X

C

p rl;m0

a.p1/CbD2

”p  1  e0

ap 

”1 i0

bp

! p ”p  1 1 X p D C pi 0 .1/ ” p   pi 0  D1 p2 p

C

X

a.p1/CbD2

p

sl

 p.p1/i 0



”p  1 j 0

p a 

”1 0

p b

is in H  . 0 p   Now  pi †D1 p .1/ ” p 2 H  , and all of the terms in the second summation above with l D a.p  1/ C b  p are in H  . Thus sl ”p  1 X p b  C 0 0 ..”  1/ / 2 H : pi p.p1/i   p1

bD2

p

8.1 The Valuation Condition

147

Since this quantity is in K CO p2 \ H  D H.j 0 /, and since b D 0; : : : ; p  1, is an R-basis for H.j 0 /, we conclude that pi 0  j 0 .

n

” p 1 0 j

b o , u t

We next consider the case i < e=p. Lemma 8.1.3. Let H be an R-Hopf order in KCp2 that induces the short exact sequence (8.2). If i < e=p, then j 0  e=p.  Proof. Since H.i  / injects into H , hthep Larson iorder H.i; 0/  H . Thus H  ” 1 ”u1  0 0 1 0 0 1 H.i; 0/ D A e ; i ; p2 D R  e0 ;  i 0 . Note that A e ; i ; p2 is not h p i , Larson since i < e=p implies i 0 > e 0 =p. By Lemma 7.1.6, u is a unit in R ” 1 e0  h p i 1 and thus A e 0 ; i 0 ; p1 ; ”u D R ” 1 . 0 2 e0 i

Let h D

”  u1 ”p  1 , k D . Then 0 e i0  Y D k; k 2 ; : : : ; k p1 ; hk; hk 2 ; : : : ; hk p1 ; : : : ;

hp1 k; hp1 k 2 ; : : : ; hp1 k p1 ; 1; h; : : : ; hp1



 is an R-basis for A.e 0 ; i 0 ; p1 is given by the 2 /. Therefore an R-basis f˛m g for H matrix product   Y M D ˛1 ; ˛2 ; : : : ; ˛p2 ;

where M is an upper-triangular p 2  p 2 matrix with entries rl;m 2 R. The condition i 0 > e 0 =p implies pi 0 > e 0  j 0 , and so H  is not a Larson order, ”1 yet it does contain a largest Larson order A.„.H  //. Thus H  contains for t some t  0. There exist elements sm 2 R with 2

p X ”1 D sm ˛m ; t mD1

and so there exist elements ql 2 R for which ”1 D t

p 2 p

X

2

ql h k C a b

lDa.p1/CbD1

p X

q hc ;

c D p 2  p C 1:

Dc

One has ql D 0 for 2  l  p 2  p, and so p2

X ”1 D q1 k C q hc ; t  Dc

c D p2  p C 1 p2

”1 1  u1 X c C q h ; D q1 i 0 C q1  i0 Dc

c D p 2  p C 1:

8 Hopf Orders in KCp 2

148

If ord.q1 / > i 0 , then

”1 t

62 H  , and thus ord.q1 /  i 0 . Moreover, 2

p X

q hc 2 H  \ KCp2 D H.j 0 /;

Dc 1u1 i0

2 H.j 0 /. Consequently, 1  u1 2 H.j 0 /. Now, by Lemma 7.1.6, and so q1  t u ord.1  p2 / D e 0 =p  j ; that is, j 0  e=p. We now show that H in KCp2 satisfies the valuation condition for n D 2. Proposition 8.1.2. Let H be an R-Hopf order in KCp2 that induces the short exact sequence of (8.2). Then either pj  i or pi 0  j 0 . Proof. Suppose i  e=p. Then, by Lemma 8.1.2, pj 0  i 0 . On the other hand, if i < e=p, then j 0  e=p by Lemma 8.1.3, and thus Lemma 8.1.2 can be applied to the dual short exact sequence to obtain pj  i . t u

8.2 Some Cohomology In this section, we give a review of cohomology of groups, which will be needed in the subsequent section. Let Cm denote the cyclic group of order m, generated by , let G be an Abelian group, and let G m denote the subgroup of G generated by fg m W g 2 Gg. An extension of G by Cm is a short exact sequence of groups i

s

E W 1 ! G ! G 0 ! Cm ! 1: Two extensions E1 , E2 are equivalent if there exists an isomorphism G10 ! G20 such that the diagram E1 W 1 ! G ! G10 ! Cm ! 1 jj # jj 0 E 2 W 1 ! G ! G2 ! C m ! 1

(8.4)

commutes. Let E.G; Cm / denote the collection of equivalence classes of extensions of G by Cm . In this section, we compute E.G; Cm / following [Rot02, 10.3]. A cocycle is a function f W Cm  Cm ! G that satisfies the conditions f .1; j / D f . i ; 1/ D 1

(8.5)

        f i ; j f i Cj ; k D f j ; k f i ; j Ck

(8.6)

and

for i; j; k D 0; : : : ; m  1.

8.2 Some Cohomology

149

Proposition 8.2.1. The collection of cocycles, denoted by C.G; Cm /, is an Abelian group under the product .f g/. i ; j / D f . i ; j /g. i ; j /; with identity element f . i ; j / D 1, for all i; j . t u

Proof. Exercise.

We can identify a subgroup of C.G; Cm / as follows. Let h W Cm ! G be a function with h.1/ D 1. Let fh W Cm  Cm ! G be defined as fh . i ; j / D h. i /.h. i Cj //1 h. j / for i; j D 0; : : : ; m  1. Then it is routine to check that fh is a cocycle. We write the cocycle fh as @h and call it the coboundary of h. Let B.G; Cm / denote the collection of coboundaries B.G; Cm / D f@h W h W Cm ! G; h.1/ D 1g: Then B.G; Cm / is a subgroup of C.G; Cm /. Proposition 8.2.2. The quotient group C.G; Cm /=B.G; Cm / is in a 1-1 correspondence with the equivalence classes of extensions E.G; Cm /. Proof. Let f1 ; f2 be cocycles that satisfy the condition 1   1    D @h i ; j D h. i / h. i Cj / h. j / f1 i ; j f2 . i ; j / for some @h 2 B.G; Cm /. Then h. i Cj /f1 . i ; j / D h. i /h. j /f2 . i ; j /: On the Cartesian product G10 D G  Cm , define a multiplication .a; i / f1 .b; j / D .abf1 . i ; j /; i Cj /; and on G20 D G  Cm define a multiplication .a; i / f2 .b; j / D .abf2 . i ; j /; i Cj /: Then there exist extensions of groups E1 W 1 ! G ! G10 ! Cm ! 1 and

E2 W 1 ! G ! G20 ! Cm ! 1:

(8.7)

8 Hopf Orders in KCp 2

150

We claim that E1 and E2 are equivalent. To this end, define a map h W G10 ! G20 by the rule h..a; i // D .ah. i /; i /. Then h..a; i / f1 .b; j // D h..abf1 . i ; j /; i Cj // D .abh. i Cj /f1 . i ; j /; i Cj / D .abh. i /h. j /f2 . i ; j /; i Cj / by (8.7) D .ah. i /; i / f2 .bh. j /; j / D h..a; i // f2 h..b; j //; and so h is an isomorphism that makes the diagram (8.4) commute. Consequently, there is a well-defined map ‰ W C.G; Cm /=B.G; Cm / ! E.G; Cm /; where ‰.f B.G; Cm // is the equivalence class represented by E W 1 ! G ! G 0 ! Cm ! 1; where G 0 D G  Cm is endowed with the binary operation .a; i / f .b; j / D .abf . i ; j /; i Cj /:

(8.8) t u

Evidently, ‰ is a bijection.

The values of a given cocycle f W Cm  Cm ! G can be arranged in an m  m matrix Mf whose i; j th entry is f . i ; j / D ai;j for i; j D 0; : : : ; m  1. Note that the entries in the first row and first column are all 1 by cocycle property (8.5). Moreover, since the binary operation on G  Cm is Abelian, .a; i /  .b; j / D .b; j /  .a; i /; and so

.abf . i ; j /; i Cj / D .baf . j ; i /; j Ci /;

which implies that f . i ; j / D f . j ; i / for all i; j D 0; : : : ; m  1. Thus Mf is symmetric. We prove the following. Proposition 8.2.3. Let Cm denote the cyclic group of order m generated by , let G be an Abelian group, and let G m denote the subgroup of G generated by fgm W g 2 Gg. Then there is a group isomorphism ˆ W G=G m ! C.G; Cm /=B.G; Cm /

8.2 Some Cohomology

151

defined as ˆ.wG m / D fw B.G; Cm /; where fw W Cm  Cm ! G is the cocycle in C.G; Cm / whose values are given by the matrix 1 0 1 1 1  1 B1 1  1 wC C B B1    1 w wC Mfw D B C: B: :: :: C @ :: : :A 1

w

w



w

Proof. We first show that ˆ is well-defined on cosets of G=G m . Suppose w 2 vG m . Then wv1 D g m for some g 2 G, and the cocycle fwv1 W Cm  Cm ! G satisfies fwv1 D @h, where h W Cm ! G is defined by h. i / D g i . Thus fwv1 2 B.G; Cm /, and so ˆ is well-defined. Now, ˆ.vG m wG m / D ˆ.vwG m / D fvw B.G; Cm / D fv B.G; Cm /fw B.G; Cm /; and so ˆ is a group homomorphism. Next, suppose that ˆ.wG m / D ˆ.vG m /. Then fwv1 2 B.G; Cm /. Thus there exists a function h W Cm ! G, h.1/ D 1 for which @h D fwv1 . Thus wv1 D g m for some g 2 G, and so wG m D vG m , which shows that ˆ is an injection. To show that ˆ is surjective, let f B.G; Cm / be a coset in C.G; Cm /=B.G; Cm /. Assuming that this coset can be written in the form fw B.G; Cm / for some w 2 G, we have ‰.wG m / D f B.G; Cm /, and so ˆ is surjective. So the remainder of this proof will be concerned with proving that the coset f B.G; Cm / can be written in this form. We know that f has matrix 0

1 B1 B B B1 B Mf D B :: B: B B :: @: 1

1 a1;1 a2;1 :: : :: :

1 a1;2 a2;2 :: : :: :

  

am1;1

am1;2



1 a1;m1 a2;m1 :: : :: :

1 C C C C C C C C C A

am1;m1

for elements ai;j 2 G with Mf symmetric. In the second row of Mf , let l be the smallest integer for which a1;l 6D 1. Define a function h W Cm ! G by

8 Hopf Orders in KCp 2

152

h. / D i

1 a1;l

for i D 0; : : : ; l for i D l C 1; : : : ; m  1.

Then @h is a coboundary with matrix M@h whose first and second rows and columns, up to and including the codiagonal, have the form 0

M@h

1 B1 B B: B :: B B1 B B 1 DB B B1 B B :: B: B B :: @: 1

1 1 :: : 1 1 a1;l 1 :: :

 

1 1

1 1 a1;l

1 1

 

 1

   

   

   

   

   

   













1 

 

 

 

 

 

 

1 1 C C C C C C C C C : C C C C C C C A 

1 is in the .1; l/th and .l; 1/th places. It follows that f is congruent Here the entry a1;l modulo B.G; Cm / to a cocycle whose matrix is of the form

0

1 B1 B B B1 B B1 Mf D B B B1 B B :: @: 1

1 1 1 1 :: :

1 1  

1 1  

1   

 1  









1 

 

 

 

 

1 1 C C C C C C C: C C C C A 

Consequently, we can assume, without loss of generality, that the matrix Mf of the cocycle f is in the form above. By the cocycle condition (8.6), f . ; /f . 2 ; k / D f . ; k /f . ; 1Ck / for k, 0  k  m1. But f . ; / D f . ; k / D f . ; 1Ck / D 1 for 1  k  m3, and so f . 2 ; k / D 1 for k D 1; : : : ; m  3. Now, again by (8.6), f . 2 ; /f . 3 ; k / D f . ; k /f . 2 ; 1Ck /

8.2 Some Cohomology

153

for all k. But, as we have seen above, f . 2 ; / D f . 2 ; k / D f . 2 ; 1Ck / D 1 for 1  k  m  4. Thus f . 3 ; k / D 1 for k D 1; : : : ; m  4. Continuing in this manner, we see that f . m2 ; k / D 1 for k D 1; : : : ; m  .m  2 C 1/ D 1. It follows that the matrix for f is of the form 0 1 B1 B B B1 B Mf D B B1 B: B: @: 1

1 1 1 :: :

1 1 

1 

 

1  1

 1 





 

 

1 1 C C C C C C: C C C A 

But what about the matrix entries below the main codiagonal? We show that they are all equal. By (8.6), f . ; /f . 2 ; m1 / D f . ; 1/f . ; m1 /: But, as we have seen, this implies that f . 2 ; m1 / D f . ; m1 /: Moreover, f . ; 2 /f . 3 ; m1 / D f . ; /f . 2 ; m1 /; and so f . 3 ; m1 / D f . 2 ; m1 /: Hence f . kC1 ; m1 / D f . k ; m1 / for k D 1; : : : ; m  2. By (8.6), f . ; k /f . kC1 ; m2 / D f . ; m2Ck /f . k ; m2 / for all k. Since f . ; k / D f . ; m2Ck / D 1 for k D 2; : : : ; m  2, f . kC1 ; m2 / D f . k ; m2 / for k D 2; : : : ; m  2. Continuing in this manner, we see that the entries below the main codiagonal in a column are equal.

8 Hopf Orders in KCp 2

154

We next show that all of the entries below the main codiagonal are equal to a common value. By (8.6), f . ; /f . 2 ; m2 / D f . ; m1 /f . ; m2 / for all k, and thus f . 2 ; m2 / D f . ; m1 /: By (8.6), f . ; 2 /f . 3 ; m3 / D f . ; m1 /f . 2 ; m3 /; and thus f . 3 ; m3 / D f . ; m1 /: Continuing in this manner, we conclude that f . k ; mk / D f . ; m1 / for k D 2; : : : ; m  1. Now f . i ; j / D f . l ; k / for all i C j > m  1, l C k > m  1, and so all of the entries below the main codiagonal are equal to a common value, say w. Thus the matrix of f is in the form 0 1 B1 B B1 B B: @ ::

1 1 

1  1 :: :

 1 w

1

w

w



1 1 wC C wC C: :: C :A w

This completes the proof of the proposition.

t u

We summarize with the following proposition. Proposition 8.2.4. There is a 1-1 correspondence between the group G=G m and the equivalence classes in E.G; Cm /. Specifically, a coset wG m 2 G=G m corresponds to an equivalence class of extensions represented by Ew W 1 ! G ! Gw0 ! Cm ! 1; where the group operation on the set Gw0 D G  Cm is given as

.a; /  .b; / D i

j

if i C j < m .ab; i Cj / i Cj m / if i C j  m. .abw;

8.3 Greither Orders

155

Proof. By Proposition 8.2.3 and Proposition 8.2.2, there exists a 1-1 correspondence G=G m ! E.G; Cm /, where the coset wG m corresponds to the matrix Mfw . The definition of the binary operation follows from (8.8). t u

8.3 Greither Orders In this section, we compute the algebraic structure of the Hopf order H in the short exact sequence (8.2) assuming that i  pj , following closely the work of C. Greither [Gr92]. We shall employ the cohomology of the previous section in the following context. Let i; j be integers with 0  i; j  e 0 , and pj  i . Let Di be the R-group scheme represented by the Larson order  H.i / D R

 1 ; i

h i D Cp ;

and let Ej denote the functor represented by the polynomial algebra  R

 X  1 1 ; X ; j

where X is indeterminate.

1 is an R-Hopf algebra with coProposition 8.3.1. The R-algebra R X1 j ;X multiplication, counit, and coinverse maps induced from the corresponding maps of the K-Hopf algebra KŒX; X 1 . Consequently, Ej is an R-group scheme that over K appears as Gm;K D HomK-alg .KŒX; X 1 ; /. Proof. One readily computes  KŒX;X 1

X 1 j

 D

X 1 X 1 X 1 ˝1C1˝ C .X  1/ ˝ j j j

and KŒX;X 1 .X 1 / D X 1  X 1 . Moreover,    X  1 1 ; X R K.X;X 1 R j      X  1 1 X  1 1  R KŒX;X 1 R ; X ; X j j

1 1 since X; X  j1 2 R X1 : j ;X

and

t u

8 Hopf Orders in KCp 2

156

Observe that Ej .R/ D Uj .R/. An extension of Ej by Di is a short exact sequence of group schemes i

s

E W 1 ! Ej ! G ! Di ! 1I that is, an extension of Ej by Di is a sequence as above in which Di is the quotient sheaf of G by Ej (see Definition 3.3.3). Observe that if E is an extension of Ej by Di , then it is not obvious that iS

sS

E.S / W 1 ! Ej .S / ! G.S / ! Di .S / ! 1 is a short exact sequence of groups for an R-algebra S . M. Demazure and P. Gabriel [DG70, III, 6, 2.5] have provided the following proposition. Proposition 8.3.2. Let i; j be integers that satisfy 0  i  e 0 , 0  j  pe 0 . Then i

s

E W 1 ! Ej ! G ! Di ! 1 is an extension of Ej by Di if and only if iS

sS

E.S / W 1 ! Ej .S / ! G.S / ! Di .S / ! 1 is an extension of Ej .S / by Di .S / in the sense of 8.2 for each R-algebra S . In view of Proposition 8.3.2, two extensions, E1 , and E2 such are equivalent if there exists an isomorphism of group schemes G1 ! G2 such that the diagram E1 .S / W 1 ! Ej .S / ! G1 .S / ! Di .S / ! 1 jj # jj E2 .S / W 1 ! Ej .S / ! G2 .S / ! Di .S / ! 1 commutes for each R-algebra S . Let E.Ej ; Di / denote the collection of equivalence classes of extensions of Ej by Di . We seek to compute E.Ej ; Di /. In view of the cohomology already discussed, we should consider “cocycles modulo coboundaries.” But what plays the role of the cocycles C.G; Cm /? Let Di  Di denote the product of group schemes. Then, by Proposition 2.4.2, the representing algebra of Di  Di is H.i / ˝R H.i /. Definition 8.3.1. The natural transformation f W Di  Di ! Ej is a cocycle if, for each R-algebra S and x; y; z 2 Di .S / fS .x; y/.X /fS .xy; z/.X / D fS .y; z/.X /fS .x; yz/.X /; fS .x; 1/.X / D fS .1; y/.X / D 1: The collection of cocycles is denoted by C.Ej ; Di /.

8.3 Greither Orders

157

Analogous to Proposition 8.2.1, C.Ej ; Di / is a group under the binary operation .fS gS /.x; y/.X / D fS .x; y/.X /gS .x; y/.X / for each R-algebra S , x; y 2 Di .S /. We seek to characterize these cocycles in terms of the cocycles we’ve already developed. By Yoneda’s Lemma, f 2 C.Ej ; Di / corresponds to an R-algebra homomorphism, 

f W R

 X  1 1 ; X ! H.i / ˝R H.i /; j

that is determined by the two conditions X 7!

p1 p1 X X

am;n .em ˝ en / 2 U.H.i / ˝ H.i //

mD0 nD0

and P p1 Pp1 mD0

nD0 am;n .em

˝ en /  1

j

2 H.i / ˝ H.i /:

There is an injection  W H.i / ˝R H.i / !

p1 M

Rem ˝R

mD0

p1 M

Ren ;

nD0

and the map  f is an R-algebra homomorphism   f W R

 p1 p1 M M X  1 1 ; X Re ˝ Rem D RCp ˝R RCp : ! m R j mD0 mD0

Consequently, the algebra homomorphism  f corresponds to a natural transformation fQ W HomR-alg .RC  ˝R RC  ; / ! Ej : p

By Proposition 4.1.8,

HomR-alg .RCp

p

˝R RCp ; R/

D Cp  Cp , and so

fQR W Cp  Cp ! Ej .R/ D Uj .R/:

8 Hopf Orders in KCp 2

158

Note that fQR . m ; n /.X / D . m ; n /. f /.X / D . ; / m

n

p1 p1 X X

! am0 ;n0 .em0 ˝ en0 /

m0 D0 n0 D0

D am;n : In this manner, f gives rise to a function fO W Cp  Cp ! Uj .R/; defined as . m ; n / 7! am;n . We can now prove the following proposition. Proposition 8.3.3. The natural transformation f W Di  Di ! Ej is a cocycle in C.Ej ; Di / if and only if fO is a cocycle in C.Uj .R/; Cp /. Proof. Suppose that the natural transformation f W Di  Di ! Ej is a cocycle with algebra homomorphism f . Then, for all x; y; z 2 Di .R/, fR .x; y/.X /fR .xy; z/.X / D fR .y; z/.X /fR .x; yz/.X /: Consequently, for all l; m, 0  l; m  p  1, fQR . l ; m /.X /fQR . lCm ; n /.X / D fQR . m ; n /fQR . l ; mCn /.X /; so that al;m alCm;n D am;n al;mCn ; where m C n and l C m are taken modulo p. Thus fO. l ; m /fO. lCm ; n / D fO. m ; n /fO. l ; mCn /; and hence fO satisfies cocycle condition (8.6). Moreover, since f W Di  Di ! Ej is a cocycle, fR .x; 1/.X / D 1 D fR .1; y/.X / for all x; y 2 Di .R/. Thus, for all 0  l; m  p  1, fQR . l ; 1/.X / D 1 D fQR .1; m /.X /;

8.3 Greither Orders

159

so that al;0 D 1 D a0;m for all 0  l; m  p  1. Thus fO satisfies the cocycle condition (8.5). It follows that fO is in C.G; Cm / with G D Uj .R/, Cm D Cp . Now suppose that fO W Cp  Cp ! Uj .R/ is a cocycle obtained from the natural transformation f W Di  Di ! Ej . Then, for all l; m; n, 0  l; m; n  p  1, one has al;m alCm;n D al;mCn am;n ; where m C n and l C m are taken modulo p. Thus, p1 p1 p1 X XX

al;m alCm;n .el ˝ em ˝ en / D

lD0 mD0 nD0

p1 p1 p1 X XX

al;mCn am;n .el ˝ em ˝ en /;

lD0 mD0 nD0

which yields p1 p1 X X

! al;m .el ˝ em ˝ 1/

lD0 mD0

D

p1 p1 X X

! ak;n ..ek / ˝ en /

kD0 nD0

p1 p1 X X

! am;n .1 ˝ em ˝ en /

mD0 nD0

p1 p1 X X

! al;k .el ˝ .ek // :

lD0 kD0

Thus, for x; y; z 2 Di .S /, 0

p1 p1 XX

.x ˝ y ˝ z/ @

1

0

al;m .el ˝ em ˝ 1/A .x ˝ y ˝ z/ @

lD0 mD0

0 D .x ˝ y ˝ z/ @

p1 p1 XX mD0 nD0

XX

1

p1 p1

ak;n ..ek / ˝ en /A

kD0 nD0

1

0

am;n .1 ˝ em ˝ en /A .x ˝ y ˝ z/ @

XX

1

p1 p1

al;k .el ˝ .ek //A;

lD0 kD0

which implies .x; y/ f .X /.xy; z/ f .X / D .y; z/ f .X /.x; yz/ f .X /: Thus fS .x; y/.X /fS .xy; z/.X / D fS .y; z/.X /fS .x; yz/.X /:

8 Hopf Orders in KCp 2

160

Now, assume the condition 1 D am;0 for m D 0; : : : ; p  1, and let x 2 Di .S /. Then ! p1 X am;0 em 1Dx mD0

D .x ˝ S H.i / /

p1 p1 X X

! am;n .em ˝ en /

mD0 nD0

D .x; 1/ f .X / D fS .x; 1/.X /: In a similar manner, the condition 1 D a0;n , for n D 0; : : : ; p  1, yields fS .1; y/.X / D 1 for y 2 Di .S /. Consequently, f is a cocycle. t u We identify the group C.Ej ; Di / with the subgroup C of C.Uj .R/; Cp / defined as C D fg 2 C.Uj .R/; Cp / W g D fO for some f 2 C.Ej ; Di /g: We next consider coboundaries. Let h W Di ! Ej be a natural transformation that satisfies hS .1/ D 1, 8S . Let fh W Di  Di ! Ej be the natural transformation defined as .fh /S .x; y/.X / D hS .x/.X /.hS .xy//1 .X /hS .y/.X /: Then fh satisfies the cocycle conditions of Definition 8.3.1. We denote this cocycle by @h. The collection of all cocycles of the form f@h W h W Di ! Ej is a natural transformation, hS .1/ D 1; 8S g is a subgroup of C.Ej ; Di / denoted by B.Ej ; Di /. B.Ej ; Di / is the collection of coboundaries. Observe that the hat operation on cocycles can be applied to the natural transformation h W Di ! Ej . By Yoneda’s Lemma, h corresponds to an R-algebra homomorphism,   X  1 1 ; X ! H.i /;

h W R j which is determined by the conditions X 7!

p1 X mD0

am em 2 U.H.i //

8.3 Greither Orders

161

and .

Pp1

mD0 am em / j

1

2 H.i /:

There is an injection  W H.i / !

p1 M

Rem ;

mD0

and the map  h is an R-algebra homomorphism  p1 M X  1 1 !  h W R ;X Rem D RCp j 

mD0

that corresponds to a natural transformation hQ W HomR-alg .RCp ; / ! Ej with

hQ R W HomR-alg .RCp ; R/ D Cp ! Ej .R/ D Uj .R/:

Note that hQ R . m /.X / D . m /. h /.X / D . / m

p1 X

!

am0 em0

m0 D0

D am : In this manner, h gives rise to a function hO W Cp ! Uj .R/; defined as m 7! am . O which is One can compute the coboundary of hO (as defined in 8.2) to obtain @h, in B.Uj .R/; Cp /. Proposition 8.3.4. The cocycle f W Di  Di ! Ej is a coboundary (f D @h, for O some h) if and only if the cocycle fO is a coboundary (fO D @h). Proof.  Let h W Di ! Ej be a natural transformation with algebra map h W Pp1 Pp1 X  1 1 R ;X ! H.i / given by X 7! mD0 am em 2 U.H.i //, mD0 am em 2 j 1 C  j H.i /.

8 Hopf Orders in KCp 2

162

Let f 2 C.Ej ; Di / be a cocycle that satisfies fR .x; y/.X / D hR .x/.X /.hR .xy/.X //1 hR .y/.X / for all x; y 2 Di .R/. Then, for 0  m; n  p  1, fQR . m ; n /.X / D hQ R . m /.X /.hQ R . mCn /.X //1 hQ R . n /.X /: Thus am;n D am .amCn /1 am O for all 0  m; n  p  1 (m C n taken modulo p), and so fO D @h. O O Now suppose @h D f ; that is, suppose that am;n D am .amCn /1 an for 0  m; n  p  1 (m C n taken modulo p). Then p1 X

am;n .em ˝ en / D

m;nD0

p1 X

am .amCn /1 an .em ˝ en /

m;nD0

D

p1 X

am

mD0

D

p1 X mD0

D

p1 X mD0

p1 X

an

nD0

am

p1 X nD0

p1 X

ak1

kD0

an

p1 X

lD0

ak1

kD0

am .em ˝ 1/

p1 1 X lk mlCnl p .p /.em ˝ en / p

p1 X kD0

p1 1 X lk l p . ˝ l /.em ˝ en / p lD0

ak1 H.i / .ek /

p1 X

an .1 ˝ en /:

nD0

(8.9) Let f denote the algebra map of f . Let x; y 2 Di .S /. Then the relation (8.9) implies that .x; y/ f .X / D .x/ h .X /..xy/ h .X //1 .y/ h .X /; which yields fS .x; y/.X / D hS .x/.X /.hS .xy/.X //1 hS .y/.X /; and hence @h D f .

t u

8.3 Greither Orders

163

By Proposition 8.3.4, we can identify the group B.Ej ; Di /, with the subgroup of B.Uj .R/; Cp / defined as B D f@k W k D hO for some h W Di ! Ej g: Analogous to Proposition 8.2.2, there is a bijection ‰ W C =B ! E.Ej ; Di /; and through ‰ one can define a group operation on E.Ej ; Di /: for E1 ; E2 , let E1 E2 D ‰.xy/; where ‰.x/ D E1 and ‰.y/ D E2 . Clearly, E.Ej ; Di / Š C =B. We want to characterize the quotient C =B. p

Proposition 8.3.5. C =B Š Upi 0 Cj .R/=Ui 0 Cj .R/. Proof. Let fO 2 C . By 8.2, fO has matrix 0

1 B1 B B B1 B MfO D B :: B: B B :: @: 1

1 a1;1 a2;1 :: : :: :

1 a1;2 a2;2 :: : :: :

  

am1;1

am1;2



1 a1;m1 a2;m1 :: : :: :

1 C C C C C C C C C A

am1;m1

for elements ai;j 2 Uj .R/ with MfO symmetric. Since p1 p1 X X

am;n .em ˝ en / 2 1 C  j .H.i / ˝ H.i //;

mD0 nD0

* .”  1/ ˝ .”  1/ ; a

b

p1 p1 X X

+ 0

am;n .em ˝ en / 2  .aCb/i R

mD0 nD0

for 0  a; b  p  1, a C b > 0. In the second row of MfO , let l be the smallest integer for which a1;l 6D 1. Now, * ”  1 ˝ .”  1/ ; l

p1 p1 X X mD0 nD0

+ 0

am;n .em ˝ en / D 1  a1;l 2  .1Cl/i R: (8.10)

8 Hopf Orders in KCp 2

164

P Consider the partial sum lnD0 a1;n en . Then, by (8.10), this sum satisfies the first l C2 conditions for membership in H.i / (see Lemma 7.1.6). By using Lemma 7.1.7, we can extend this sum to an element y 2 H.i /. The element y determines an R-algebra map 

 X  1 1

y W R ;X ! H.i /; j

X 7! y;

which corresponds to a natural transformation s D h y W Di ! Ej and a function sO W Cp ! Uj .R/: Now @Os is a cocycle in C , and fO  @Os has a matrix whose second row satisfies a1;0 D a2;0 D    D a1;l D 1: As in 8.2, we see that fO  @hO D fw for some @hO 2 B, where fw has p  p matrix 0

1 B1 B B M w D B1 B: @ ::

1 1 

1  1 :: :

 1 w

1

w

w



1 1 wC C wC C :: C :A w

for w 2 Uj .R/ and so each element fO 2 C is congruent modulo B to a cocycle in C of the form fw . Next, one shows that a cocycle fw in C.Uj .R/; Cp / with matrix in the form Mw , w 2 Uj .R/, corresponds to a cocycle in C if and only if ord.1  w/  pi 0 C j ; that is, if and only if w 2 Upi 0 Cj .R/. Moreover, fw is trivial in C if and only if p w 2 Ui 0 Cj .R/. (See Greither’s proof in [Gr92] for details.) It follows that C =B Š p t u Upi 0 Cj .R/=Ui 0Cj .R/. As a consequence of Proposition 8.3.5, p

E.Ej ; Di / Š Upi 0 Cj .R/=Ui 0Cj .R/: Let Egen-triv .Ej ; Di / denote the collection of generically trivial extensions; that is, those elements in E.Ej ; Di / that over K appear as 1 ! Gm;K ! Gm;K  p;K ! p;K ! 1: Proposition 8.3.6. Egentriv .Ej ; Di / Š Ui 0 C.j=p/ .R/=Ui 0 Cj .R/.

8.3 Greither Orders

165

Proof. Over K, one has an isomorphism E.Gm;K ; p;K / Š K  =.K /p ; and so the generically trivial extensions correspond to units w that are pth roots in K. t u h i 1 p Let Dj D HomR-alg .H.j /; / with H.j / D R  j ,  D 1, and let E.Dj ; Di / denote the extensions of Dj by Di . Let Egen-triv .Dj ; Di / denote the collection of extensions that over K appear as 1 ! p;K ! p;K  p;K ! p;K ! 1: We want to replace Ej with Dj in Proposition 8.3.6 and compute Egen-triv .Dj ; Di /. The group schemes Ej and Di are related, and so this computation is possible. Let G0m;K D HomK-alg .KŒY; Y 1 ; /; with Y indeterminate, and let Epj D HomR-alg .R a short exact sequence of R-group schemes

Y 1  pj

; Y 1 ; /. Then there exists

p

1 ! Di ! Ej ! Epj ! 1

(8.11)

that is induced from the short exact sequence of K-group schemes p

1 ! p;K ! Gm;K ! G0m;K ! 1 of 3.4. We now state C. Greither’s main result [Gr92, II, Corollary 3.6(b)]. Proposition 8.3.7. (Greither) Let i; j be integers with 0  i; j  e 0 , pj  i . (i) There is an isomorphism Egentriv .Dj ; Di / Š .Ui 0 C.j=p/ .R/ \ U.i 0 =p/Cj .R//=Ui 0 Cj .R/; where the coset vUi 0 Cj .R/ corresponds to an equivalence class of extensions represented by the extension Ev W 1 ! Dj ! G ! Di ! 1: (ii) The extension Ev corresponds to a short exact sequence of R-Hopf orders Ev0



  1 av1  1 W R ! H.i / ! R ; ! H.j / ! R; i j

where av1 D

p1 X mD0

vm em , where em are the minimal idempotents in KCp D

Kh i. The middle term is an R-Hopf order in K.Cp  Cp /.

8 Hopf Orders in KCp 2

166

Proof. Proof of (i). The short exact sequence (8.11) induces the exact sequence p

1 ! Egen-triv .Dj ; Di / ! Egen-triv .Ej ; Di / ! Egen-triv .Epj ; Di /: Consequently,  p Egen-triv .Dj ; Di / D ker Egen-triv .Ej ; Di / ! Egen-triv .Epj ; Di / : Now, by Proposition 8.3.6, Egen-triv .Ej ; Di / Š Ui 0 C.j=p/ .R/=Ui 0 Cj .R/ and Egen-triv .Epj ; Di / Š Ui 0 Cj .R/=Ui 0 Cpj .R/; from which (i) follows. We next prove (ii). Let Ev be the generically trivial extension of Dj by Dj determined by the element v 2 Ui 0 C.j=p/ .R/ \ U.i 0 =p/Cj .R/. Then, over K, Ev appears as Ev;K W 1 ! HomK-alg .H.j / ˝R K; / ! HomK-alg .H.j / ˝R K; /  HomK-alg .H.i / ˝R K; / ! HomK-alg .H.i / ˝R K; / ! 1: The group structure of HomK-alg .H.j / ˝R K; /  HomK-alg .H.i / ˝R K; / is given by a cocycle fO 2 C whose matrix is Mvp . The extension Ev;K is equivalent to the trivial extension 1 ! p;K ! p;K  p;K ! p;K ! 1; and so the natural transformation h W Di ! Ej given as X 7! determines a group scheme isomorphism

Pp1 mD0

vm em

h W HomK-alg ..H.j / ˝R H.i // ˝R K; / ! HomK-alg .Khi ˝ Kh i; /; defined as hS .x; y/ D .xhS .y/; y/, for a K-algebra S and x 2 HomK-alg .H.j / ˝R K; S /, y 2 HomK-alg .H.i / ˝R K; S /.

8.3 Greither Orders

167



 1 1 Note that Khi ˝ Kh i Š KŒ; and .H.j / ˝R H.i // Š R ; . j i Thus the isomorphism h yields the K-algebra homomorphism  % W KŒ; ! K

1 1 ; j i



with %./ D av1 , %. / D . Thus       1 av1  1 1 1 ; ; DR ; % R j i i j which is thei representing algebra of the group scheme G. Consequently, h 1 av 1 is an R-Hopf algebra. R i ; j h i av1 1 We leave it to the reader to show that R 1 is an R-Hopf order in i ; j K.Cp  Cp / that induces the short exact sequence   1 av1  1 ; ! H.j / ! R: R ! H.i / ! R i j 

t u

But how do we obtain Hopf orders in KCp2 from the extensions of Proposition 8.3.7? The key is to endow the collection of short exact sequences of Hopf orders with a group product, which we describe as follows. Let s1

E .1/ W R ! H.i / ! H1 ! H.j / ! R; s2

E .2/ W R ! H.i / ! H2 ! H.j / ! R; be short exact sequences of Hopf algebras. Recalling that the tensor product of two Hopf algebras is again a Hopf algebra, we obtain a short exact sequence of R-Hopf algebras, s1 ˝s2

R ! H.i / ˝R H.i / ! H1 ˝R H2 ! H.j / ˝R H.j / ! R: There is a unique map that makes the following diagram commute: coker.H.j / / D " R ! H.i / ˝R H.i / ! H1 ˝R H2 k k " R ! H.i / ˝R H.i / ! ker. /

s1 ˝s2

coker.H.j / / "

! H.j / ˝R H.j / ! R k " H.j / ! H.j / !R

8 Hopf Orders in KCp 2

168

And there is a map

that makes the following diagram commute:

R ! H.i / ˝R H.i / ! k #m R!

ker. / #

! H.j / ! R k k

! ker. /=ker.m/ ! H.j / ! R

H.i /

The short exact sequence in the bottom row is the Baer product E D E .1/ E .2/ : The Baer product endows the collection of short exact sequences with the structure of a group. Now, in our case, since pj  i , there exists an extension of Larson orders 

 1 1 E0 W R ! H.i / ! R ; j ! H.j / ! R; p D : i  The Baer product E0 Ev0 is an extension 

 gp  1 gav1  1 ; R ! H.i / ! R ! H.j / ! R; i j where g D , gp D p D , whose middle term is an R-Hopf order in KCp2 that we call a Greither order. A Greither order is determined by two valuation parameters i and j and one unit parameter u D v1 , and will be denoted by A.i; j; u/. Now suppose that H is an R-Hopf order in KCp2 , that induces the short exact sequence E W R ! H.i / ! H ! H.j / ! R; where pj  i . Then the Baer product E01 E is a short exact sequence of the form R ! H.i / ! H 0 ! H.j / ! R; where H 0 is an R-Hopf order in K.Cp  Cp /. There is a corresponding short exact sequence of R-group schemes 1 ! Dj ! HomR-alg .H 0 ; / ! Di ! 1; which is genericaly trivial. Now, by Proposition 8.3.7, E01 E D Ev0 for some v 2 Ui 0 C.j=p/ .R/ \ U.i 0=p/Cj .R/. Thus E D E0 Ev0 , and so, H is a Greither order. Thus we have solved the problem that was stated at the beginning of this section: for pj  i , the middle term in the short exact sequence R ! H.i / ! H ! H.j / ! R

8.3 Greither Orders

169

is a Greither order A.i; j; u/, where u represents a class in the quotient .Ui 0 C.j=p/ .R/ \ U.i 0 =p/Cj .R//=Ui 0 Cj .R/. If j 0  pi 0 , then Greither’s result may be used to give the structure of H  . How does the collection of Greither orders relate to the Larson orders in KCp2 ? It is not too hard to prove the following. Proposition 8.3.8. Let A.i; j; u/ be a Greither order. Then A.i; j; u/ is the Larson order H.i; j / in KCp2 if and only if ord.1  u/  i 0 C j . t u

Proof. Exercise.

Not all Hopf orders in KCp2 are Greither orders, however. For example, in the short exact sequence R ! H.e 0 / ! RCp2 ! H.e 0 / ! R; one has e 0 6 pe 0 . But in view of the valuation condition (Proposition 8.1.2), a given R-Hopf order in KCp2 is either of the form A.i; j; u/ or A.i; j; u/ . So, to give a full account of the structure of Hopf orders in KCp2 , one needs to obtain the algebraic structure of A.i; j; u/ . We need a lemma. Let hgi D Cp2 , h”i D CO p2 , and let h ; i W K CO p2 KCp2 7! K denote the duality map. Lemma 8.3.1. Let ei denote the minimal idempotents of KCp , and let eOj denote the minimal idempotents of K CO p . Then .paCb/.pcCd /

heOk ” pcCd ; ej g paCb i D p2 if j D d and k D b, and is 0 otherwise.

t u

Proof. Exercise. Proposition 8.3.9. Assume that K contains p2 , and let  A.i; j; u/ D R

g p  1 gau  1 ; i j

be a Greither order in KCp2 . Let B D R  J DB 1 where uQ D p1 , auQ D 2 u A.i; j; u/ .

Pp1

Q mD0 u

m

h

” p 1 0 j



i , and let

 ”auQ  1 ; i0

m , and m D

1 p

Pp1 lD0

pm ” pl . Then J D

8 Hopf Orders in KCp 2

170

Proof. By Proposition 7.1.3, it suffices to show that hJ; A.i; j; u/i  R; and this is equivalent to the conditions 0

h” p  1; .g p  1/q .gau  1/r i 2  qi Crj Cj R

(8.12)

for q; r D 0; : : : ; p  1, q C r > 0, and 0

h”auQ  1; .g p  1/q .gau  1/r i 2  qi Crj Ci R

(8.13)

for q; r D 0; : : : ; p  1, q C r > 0. One quickly sees that (8.12) is equivalent to 0 .p  1/r 2  rj Cj R for r D 1; : : : ; p  1, which holds. q;r X p q r To show that (8.13) holds, let S D h”auQ  1; .g  1/ .gau  1/ i, let c;d D0

denote

q r X X

, and let C.c; d / D

q  r  c

d

.1/qc .1/rd . Then

cD0 d D0

SD

q;r X

C.c; d /h”auQ  1; gpc .gau /d i

c;d D0

D

q;r X

C.c; d /h”auQ ; gpc .gau /d i 

c;d D0

D

q;r X

D

C.c; d /h”auQ ; gpc .gau /d i 

q;r X c;d D0

D

q;r X

C.c; d /h”auQ ; aud g

pcCd

q;r X c;d D0

C.c; d /

i

q;r X

C.c; d /

c;d D0

C.c; d /

X

ud i uQ j heOj ”; ei g pcCd i 

q;r X

C.c; d /

c;d D0

i;j

pcCd

C.c; d /ud uQ d p2



c;d D0

D

q;r X c;d D0

c;d D0

D

C.c; d /h1; g pc .gau /d i

c;d D0

c;d D0 q;r X

q;r X

q;r X

C.c; d / (by Lemma 8.3.1)

c;d D0

C.c; d /.uQup2 /d pc 

q;r X c;d D0

C.c; d /

8.4 Hopf Orders in KC4 , KC9

D

q;r X c;d D0

D

q;r X

171

C.c; d /pc 

q;r X

C.c; d /

since uQup2 D 1

c;d D0

C.c; d /.pc  1/:

c;d D0

Now S D 0 unless q  1, r D 0. In this case, S D .p  1/q , which is in R. Thus (8.13) holds. u t  qi 0 Ci

And so we have shown that an arbitrary R-Hopf order in KCp2 can be written in the form A.i; j; u/ for some integers 0  i; j  e 0 and unit u 2 R. The largest Larson order in A.i; j; u/ can be computed as follows. Proposition 8.3.10. Let A.i; j; u/ be an R-Hopf order in KCp2 . Then H.i; l/ D A.„.A.i; j; u/// is the Larson order H.i; l/, where l D j if ord.1  u/  i 0 C j and l D i  e 0 C ord.1  u/ otherwise. Proof. If ord.1  u/  i 0 C j , then v D u1 corresponds to the trivial class in Egen-triv .Dj ; Di /. Thus, the Baer product E0 Ev0 D E0 , which says that A.i; j; u/ D H.i; j /. On the other hand, suppose that ord.1  u/ < i 0 C j . We have   au  1 g1 gau  1 D g : C j j j Now, by Lemma 7.1.6, l D ord.1  u/  i 0 is the largest integer for which au 1 2 H.i /. Therefore l is the largest integer for which g1 2 H.i /. Thus, l l A.„.A.i; j; u/// D H.i; l/. t u

8.4 Hopf Orders in KC4 , KC9 In this section, we present an alternate approach to proving the valuation condition (Proposition 8.1.2) for the cases p D 2; 3. We show that if s

R ! H.i / ! H ! H.j / ! R is a short exact sequence where H is a Hopf order in KC4 or KC9 , then either pj  i or pi 0  j 0 . We begin with a lemma.

8 Hopf Orders in KCp 2

172

Lemma 8.4.1. Let H be an R-Hopf order in KCp2 . Suppose H  can be written in the form  p  ”  1 ”u  1  H DR ; ; j 0 i0 Pp1 O where u D mD0 bm m 2 K Cp and where m are the minimal idempotents in Kh” p i. Then H is of the form  g p  1 gav  1 ; ; H DR i j 

where av D in Khgp i.

Pp1

m mD0 v fm ,

1 v D p1 2 b1 , and where fm are the minimal idempotents

1 Proof. Let v D p1 2 b1 . We claim that



 gav  1  ; H  R; j

which is equivalent to ord .hgav  1; .” p  1/r .”u  1/s i/  j C rj 0 C si 0

(8.14)

for r; s D 0; : : : ; p  1, r C s > 0. Now ! s X s q q .1/sq p2 vq b1 hgav  1; .” p  1/r .”u  1/s i D .p  1/r q qD0 ! s X s r .1/sq D .p  1/ q qD0 since p2 vb1 D 1. Thus, for s  1, (8.4) holds since the sum above is 0. For s D 0, one has hgav  1; .” p  1/r .”u  1/s i D .p  1/r ; in which case (8.4) holds since r  ord.p  1/ D re 0  j C rj 0 . It follows that gav  1 2 H. j An application of Proposition 7.1.3 then shows that H is of the form claimed. t u Proposition 8.4.1. Let H be an R-Hopf order in KCp2 that induces the short exact sequence R ! H.i / ! H ! H.j / ! R. Suppose H  can be written in the form

8.4 Hopf Orders in KC4 , KC9

173



 ” p  1 ”u  1 H DR ; ; j 0 i0 

where u D

Pp1 mD0

bm m 2 Kh” p i. Then either i  pj or j 0  pi 0 .

Proof. By Lemma 8.4.1, H has the form 

gp  1 gav  1 H DR ; i j



for some unit v in R. Now, by [Ch00, 31.3], ord.p vp  1/  pj C i 0 and ord.vp  1/  j C pi 0 ; and so, by [Ch00, 31.4], either i  pj or j 0  pi 0 .

t u



So it remains to show that H can be written in the form of Lemma 8.4.1. We begin with the case p D 2. Let H be an R-Hopf order in KC4 , hgi D C4 . Then 2 H D RŒ g 1 i ; ‰ , where ‰ is an element of KC4 for which s.‰/ is the generator g1 of the Larson order H.j /. Since KC4 D Ke0 ˝ Ke1 ˝ Ke2 ˝ Ke3 , ‰ has the j form ‰ D a0 e0 C a1 e1 C a2 e2 C a3 e3 for some elements a0 ; a1 ; a2 ; a3 2 K. Let e00 , e10 be the idempotents in Khgi. Since s.‰/ D a0 e00 C a2 e20 , a0 D 0 and a2 D c D Put h D

g 2 1 . i

42 1 . j

Thus ‰ D a1 e1 C ce2 C a3 e3 .

Then an R-basis for H is f1; h; ‰; h‰g :

Now, let m be the idempotents in K CO 4 , and let ˆ D b0 0 C b1 1 C b2 2 C b3 3 be an element in K CO 4 . Let b0 D 0 and b2 D d D such that

42 1 0 . i

hH; ˆi  R: But these can be found by choosing b1 and b3 such that h‰; ˆi D 0; h.g  1/‰; ˆi D 0: 2

We seek conditions on ˆ

8 Hopf Orders in KCp 2

174

The system above corresponds to a system of equations ha1 e1 C ce2 C a3 e3 ; b1 1 C d 2 C b3 3 i D 0; ha1 e1 C a3 e3 ; b1 1 C d 2 C b3 3 i D 0; which yields the 2  2 linear system      1 b1 .a1  c C a3 /d a1 4 C c42 C a3 43 a1 43 C c42 C a3 41 ; D a1 41 C a3 43 a1 43 C a3 41 b3 .a1 C a3 /d whose solution is readily obtained as 43 x  1 ; i0

b1 D

4 x  1 ; i0

ˆD

4 x  1 42  1 43 x  1 ”ax  1  C  C 3 D : 1 2 0 0 i i i0 i0

b3 D

with x D

a3 C a1 : a3  a1

Thus

and let J D AŒ ”axi1 Now, let A D RŒ ” 1 0 . Since hH; ˆi  R, hH; J i  R. j0 An application of Proposition 7.1.3 then shows that 2



 ” 2  1 ”ax  1 H DR : 0 ; 0 j i 

Hence H  is in the form of Lemma 8.4.1. We repeat this calculation for p D 3. Put  D 9 . Let H be an R-Hopf order in 3 KC9 , C9 D hgi. Then H D RŒ g 1 i ; ‰ , where ‰ is an element of KC9 for which generates the Larson order H.j /. We can assume that ‰ has the form s.‰/ D g1 j ‰ D c0 e0 C a1 e1 C a2 e2 C c1 e3 C a4 e4 C a5 e5 C c2 e6 C a7 e7 C a8 e8 for some elements a1 ; a2 ; a4 ; a5 ; a7 ; a8 2 K, and c0 D 0, c1 D hD

g 3 1 . i

 3 1 , c2 j

D

 6 1 . j

An R-basis for H is  ˚ 1; h; h2 ; ‰; h‰; h2 ‰; ‰ 2 ; h‰ 2 ; h2 ‰ 2 :

Now, let m denote the idempotents in K CO 9 , and let ˆ D d0 0 C b1 1 C b2 2 C d1 3 C b4 4 C b5 5 C d2 6 C b7 7 C b8 8

Put

8.4 Hopf Orders in KC4 , KC9

175

3 6 be an element of K CO 9 with d0 D 0, d1 D 1 , and d2 D 1 . We seek conditions i0 i0 on ˆ such that hH; ˆi  R. But these can be found by choosing b1 ; b2 ; b4 ; b5 ; b7 ; b8 such that

h‰; ˆi D 0; h.g 3  1/‰; ˆi D 0; h.g3  1/2 ‰; ˆi D 0; h‰ 2 ; ˆi D 0; h.g3  1/‰ 2 ; ˆi D 0; h.g 3  1/2 ‰ 2 ; ˆi D 0: This system corresponds to the system of equations ha1 e1 C a2 e2 C c1 e3 C a4 e4 C a5 e5 C c2 e6 C a7 e7 C a8 e8 ; b1 1 C b2 2 C d1 3 C b4 4 C b5 5 C d2 6 C b7 7 C b8 8 i D 0I h.  1/a1 e1 C . 2  1/a2 e2 C .  1/a4 e4 C . 2  1/a5 e5 C.  1/a7 e7 C . 2  1/a8 e8 ; b1 1 C b2 2 C d1 3 C b4 4 C b5 5 C d2 6 C b7 7 C b8 8 i D 0I h.  1/2 a1 e1 C . 2  1/2 a2 e2 C .  1/2 a4 e4 C . 2  1/2 a5 e5 C.  1/2 a7 e7 C . 2  1/2 a8 e8 ; b1 1 C b2 2 C d1 3 C b4 4 C b5 5 C d2 6 C b7 7 C b8 8 i D 0I ha12 e1 C a22 e2 C c12 e3 C a42 e4 C a52 e5 C c22 e6 C a72 e7 C a82 e8 ; b1 1 C b2 2 C d1 3 C b4 4 C b5 5 C d2 6 C b7 7 C b8 8 i D 0I h.  1/a12 e1 C . 2  1/a22 e2 C .  1/a42 e4 C . 2  1/a52 e5 C.  1/a72 e7 C . 2  1/a82 e8 ; b1 1 C b2 2 C d1 3 C b4 4 C b5 5 C d2 6 C b7 7 C b8 8 i D 0I h.  1/2 a12 e1 C . 2  1/2 a22 e2 C .  1/2 a42 e4 C . 2  1/2 a52 e5 C.  1/2 a72 e7 C . 2  1/2 a82 e8 ; b1 1 C b2 2 C d1 3 C b4 4 C b5 5 C d2 6 C b7 7 C b8 8 i D 0:

8 Hopf Orders in KCp 2

176

Set A D a1  1 C a4  4 C a7  7 ;

B D a1  2 C a4  8 C a7  5 ;

C D a1 C a4 C a7 ;

E D a2  2 C a5  5 C a8  8 ;

F D a2  4 C a5  1 C a8  7 ;

G D a2 C a5 C a8 ;

0

A D a12  1 C a42  4 C a72  7 ; C 0 D a12 C a42 C a72 ; F 0 D a22  4 C a52  1 C a82  7 ; J D c1  3 C c2  6 ; J 0 D c12  3 C c22  6 ;

B 0 D a12  2 C a42  8 C a72  5 ; E 0 D a22  2 C a52  5 C a82  8 ; G 0 D a22 C a52 C a82 ; K D c1  6 C c2  3 ; K 0 D c12  6 C c22  3 :

Note that K D c1  6 C c2  3    2  2   1 1   1 D C j j D 0: Using the relation hem ; n i D  mn =9, one obtains the 6  6 linear system 0 1 0 1 . 3 C   6 G  c1  c2 /d1 C . 6 C   3 G  c1  c2 /d2 b1 B C Bb C B C . 3 d1   6 d2 /C B 2C B C B C B C 6 3 . d1   d2 /G Bb 4 C B C MB CDB C; Bb5 C B. 3 C 0   6 G 0  c1  c2 /d1 C . 6 C 0   3 G 0  c1  c2 /d2 C B C B C C @b 7 A B . 3 d1   6 d2 /C 0 @ A b8 . 6 d1   3 d2 /G 0

where M is the matrix 0 ACE CJ B CF B B A B B B E F B B 0 0 0 0 B A C E C J B C F 0 C K0 B B B A0 B0 @ E0

F0

 3 A C  6 E C J

 3 B C  6 F

 3 A

 3 B

 6 E

 6 F

 3 A0 C  6 E 0 C J 0

 3 B 0 C  6 F 0 C K 0

 3 A0

 3 B 0

 6 E 0

 6 F 0

8.4 Hopf Orders in KC4 , KC9

177

 6 A C  3 E C J

1

 6 B C  3 E

C C C C  3 F C 6 0 3 0 0C  B C E CK C C A  6 B 0

 6 A

 6 B

 3 E  6 A0 C  3 E 0 C J 0  6 A0  3 E 0

 3 F 0

This system is equivalent to the system 0

1

B A BB B B 1 B B 0 BJ B B A0 @ B0 1

0

1

0

1

0

1

 3 BA

 3

 6 BA

 6

F E

 6

F  6 E

K0

J0

K0

 3

1 F0 E0

A0 B0

 3

 6



 3

 3 FE

J0

K0

 6

6 F 0 E0

A0 B0

 3

1

3 0 i 3 C 0 i B

C C C 0 C C ; 3.c12 Cc22 / C C 0 i C  3 C 0 C A i0 B0 0

 6 0

 3 FE 0

which reduces to the system 0

1 B A  A0 B B B0 B B 0 B B B 0 B @ A00

0  3

0 F E



 3

0 F E

A0 B0



1  6 0  3 FE 0

F0 E0



 6

0

3 0 i 0 3 C 0 i B0

0



0 F E



1  3

0

 6 FE 0

3  i0 0 C 3 C  CB 0 C 0 B i C

0

0



1

0 



0 0  3 BA0  6

F0 E0

1

A B

F0 E0

1 1

B

 1

C C C; C C C A

F0 E0



 6

 1 A B



0 0 0  6 BA0  3

A0 B0



8 Hopf Orders in KCp 2

178

and ultimately to the system 0 B B B B B B B B B @

1

0

1

0

1

0

1

0

 3

0

 6

0

0 0

1 1

0 0

 6 1

0 0

 3 1

0

1

0

 3

0

 6

1

0

 6

0

 3

0



3 0 i

3 0 i

3 0 i BC 0 CB 0 0 0 AB BA

1

C C C C 0 C C: 3 C 0 i   0 C C CA AC 0 A AB 0 BA0 0

Now, the required values of b1 ; b2 ; b4 ; b5 ; b7 ; b8 satisfy 3 ; i0   3 BC 0  CB 0 3 6 b1 C  b4 C  b7 D i 0 ;  AB 0  BA0 b 1 C b4 C b7 D

b2 C  6 b5 C  3 b8 D 0; 3 ; i0   3 CA0  AC 0 3 6 ; b2 C  b5 C  b8 D i 0  AB 0  BA0 b 2 C b5 C b8 D

b1 C  6 b4 C  3 b7 D 0: Let x and y be elements of K, and set b1 D b5 D

 5 y1 0 , i

b7 D

 7 x1 0 , i

and b8 D xD

 8 y1 0 . i 1

y D  2

 

(8.15) x1 0 , i

b2 D

 2 y1 0 , i

b4 D

Then, with

BC 0  CB 0 AB 0  BA0

 ;

 CA0  C 0 A ; AB 0  BA0

the equations in (8.15) are satisfied. Now ˆD

4x  1 2y  1 3  1 x  1  C  C  C b 4 1 2 3 4 i0 i0 i0 i0 C

 5y  1 6  1  7x  1 8y  1 ”u  1  C  C  C 8 D ; 5 6 7 0 0 0 0 i i i i     i0

 4 x1 0 , i

8.5 Chapter Exercises

179

where u D 0 C x1 C y2 2 Kh” 3 i, with m the idempotents in Kh” 3 i. With this definition of ˆ, we have hH; ˆi  R: Let A D RŒ ”

3 1

j

0

, and let J D AŒ ”u1

. Since hH; ˆi  R, hH; J i  R. An i0

application of Proposition 7.1.3 then yields 

 ” 3  1 ”u  1 ; : H DR j 0 i0 

8.5 Chapter Exercises Exercises for 8.1 1. Let K be a finite extension of Q2 with ord.2/ D e. Let H be an R-Hopf order in KC4 . Prove that there is no short exact sequence of R-Hopf orders of the form R ! H.e=2/ ! H ! H.e=2/ ! R: 2. Suppose p2 2 K, let H be an R-Hopf order in KCp2 , and let R ! H.i / ! H ! H.j / ! R be a short exact sequence with pj > i . Prove that ord.1  p2 /  i 0 C .j=p/. Exercises for 8.2 3. Let h W Cm ! G be a function with h.1/ D 1. Let fh W Cm  Cm ! G be defined as fh . i ; j / D h. i /.h. i Cj //1 h. j / for i; j D 0; : : : ; m  1. Prove that fh is a cocycle. 4. Prove Proposition 8.2.1. 5. Compute all of the non-equivalent extensions in E.Z; C3 /. 6. Compute all of the non-equivalent extensions in E.Z=.p 2 /; Cp /. Exercises for 8.3 7. Prove Proposition 8.3.8. 8. Prove Lemma 8.3.1. 9. Prove that the R-Hopf orders A.i; j; v/ and A.i; j; w/ are equal if and only if ord.v  w/  i 0 C j . 10. Let A.i; j; v/ be a Greither order in KCp2 . (a) Show that A.i; j; v1 / is a Greither order in KCp2 . (b) Show that A.i; j; v/ D A.i; j; v1 / if and only if p D 2.

180

8 Hopf Orders in KCp 2

11. Let H.i; j / be a Larson order in KCp2 . Find conditions on i; j so that the linear dual H.i; j / is a Larson order in KCp2 . 12. Let A.i; j; u/ be an R-Hopf order in KCp2 . Show that ord.1  u/  i 0 C .j=p/. 13. Suppose A.i; j; v/ is a Greither order with ord.1  p2 v/  i 0 . Show that there exists an R-Hopf order of the form H.a; b/ for which H.a; b/  A.i; j; v/. p 14. By Proposition 8.3.5, C =B Š Upi 0 Cj .R/=Ui 0 Cj .R/, and by Proposition 8.2.3, p E.Uj .R/; Cp / Š Uj .R/=Uj .R/. Is C =B a subgroup of E.Uj .R/; Cp /? Exercises for 8.4 15. How would one generalize the results of 8.4 to p > 3?

Chapter 9

Hopf Orders in KCp3

In this chapter, we move on to the construction of Hopf orders in KCp3 . We assume throughout this chapter that K is a finite extension of Qp with p3 2 K. Though all Hopf orders in KCp and KCp2 are known, this is not the case for Hopf orders in KCp3 . L. Childs and R. Underwood have explored various ways to construct Hopf orders in KCp3 ; the reader is referred to the papers [CU03], [Un08b], [Un06], [UC05], and [Un96]. In the first section here, we briefly review the construction of “duality Hopf orders” of [UC05].

9.1 Duality Hopf Orders in KCp3 Let g be a generator for Cp3 , and let  be a generator for CO p3 . Let g denote the image 2 of g under the mapping KCp3 ! KCp2 , gp 7! 1, and let  denote the image of  2 under the mapping K CO p3 ! K CO p2 ,  p 7! 1. For an integer m, 0  m  e 0 , set 1 m0 D e 0  m, and, for a unit u 2 R, set uQ D p1 . Let 2 u " 2 # g p  1 gp au  1 ; A.i; j; u/ D R i j and

 A.j; k; w/ D R

g p  1 gaw  1 ; j k



be Greither orders in KCp2 . Here u; w are units in R with au D

p1 X lD0

ul

p1 p1 p1 X 1 X lq p2 q 1 X lq pq p g and aw D wl  g : p qD0 p qD0 p lD0

Moreover, pj  i and pk  j . R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 9, © Springer Science+Business Media, LLC 2011

181

9 Hopf Orders in KCp 3

182

By Proposition 8.3.9, the linear duals of these Hopf orders are Hopf orders in KCp2 of the form  p    1  auQ  1  0 0 ; ; auQ 2 Kh p i; A.i; j; u/ D A.j ; i ; uQ / D R j 0 i0 # " 2  p  1  p awQ  1 2  0 0 ; awQ 2 Kh p i: A.j; k; w/ D A.k ; j ; w/ Q DR ; 0 0 k j   We shall extend the rank p 2 Hopf order A.i; j; u/ to obtain a Hopf order of rank p 3 . To do this, we need to select a “correct” generator ‰ 2 KCp3 that maps to gaw  1 under the canonical surjection KCp3 ! KCp2 . k Pp1 l 1 Pp1 lq p2 q Let v be a unit of R, and let av D . Let fepmCn g, qD0 p g lD0 v p m; n D 0; : : : ; p  1, denote the set of minimal idempotents in Khgp i Š KCp2 , and let p1 X wm epmCn : bw D m;nD0

Note that av bw D

p1 X

vn wm epmCn :

m;nD0

Let

 gav bw  1 : H D A.i; j; k; u; v; w/ D A.i; j; u/ k Pp1 lq 2 Pp1 Also, let x be a unit of R and let ax D lD0 x l p1 qD0 p  p q . Let fpmCn g, m; n D 0; : : : ; p  1, denote the set of minimal idempotents in Kh p i Š K CO p2 , and let p1 X buQ D uQ m pmCn 

m;nD0

and



 ax buQ  1 J D A.k ; j ; i ; w; Q x; uQ / D A.k ; j ; w/ Q : i0 0

0

0

0

0

We wish to find conditions on v and x such that H and J are R-orders in KCp3 and hH; J i  R. Then one can show that disc.J / D disc.H  /. Thus, by Proposition 4.4.10, J D H  and H; J are Hopf orders. First, we find conditions for H to be an R-order. Lemma 9.1.1. The algebra "

2

g p  1 gp au  1 gav bw  1 ; ; H D A.i; j; k; u; v; w/ D R i j k

#

9.1 Duality Hopf Orders in KCp 3

183

is an R-order in KCp3 if the following inequalities hold: (i) ord.vp p2  1/  i 0 C pk; (ii) ord.Qu  1/  i 0 > 0; (iii) ord.Qu  1/ C ord.wp p  1/  e 0 C i 0 C pk. gav bw  1 satisfies a monic k polynomial of degree p with coefficients in A.i; j; u/ (see [UC05, 3]). Thus H is finitely generated over R. Thus, by [Rot02, Theorem 9.3], H is free and of finite t u rank over R. Clearly, KH D KCp3 , and so H is an R-order. Proof. The conditions (i), (ii), and (iii) imply that

Similarly, we have the following lemma. Lemma 9.1.2. The algebra

"

2

 p  1  p awQ  1 ax buQ  1 J D A.k ; j ; i ; w; Q x; uQ / D R ; ;  k0 j 0 i0 0

0

#

0

is an R-order in K CO p3 if the following inequalities hold: (i) ord.x p p2  1/  k C pi 0 ; (ii) ord.w  1/  k > 0; (iii) ord.w  1/ C ord.Qup p  1/  e 0 C k C pi 0 . ax bwQ  1 satisfies a monic k 0 0 Q (see [UC05, 3]). Thus J is free and polynomial with coefficients in A.k ; j ; w/ of finite rank over R. Since KJ D K CO p3 , J is an R-order in K CO p3 . t u Proof. The conditions (i), (ii), and (iii) imply that

With some additional conditions, we can show that J and H are dual Hopf orders in KCp3 . For a; b 2 R, let G.a; b/ denote the Gauss sum of a and b, defined as p1 p1 1 X X mn n m  a b : G.a; b/ D p mD0 nD0 p Q x; uQ / be Proposition 9.1.1. Let H DA.i; j; k; u; v; w/ and J DA.k 0 ; j 0 ; i 0 ; w; R-algebras that satisfy the hypotheses of Lemmas 9.1.1 and 9.1.2, with the additional conditions (i) (ii) (iii) (iv)

e 0 > ord.w  1/, e 0 > ord.Qu  1/, ord.w  1/ C ord.Qu  1/  e 0 C . p1 /.i 0 C k C e 0 /, and p vxp3 G.Qu; w/ D 1.

Then H and J are Hopf orders in KCp3 with J D H  . Proof. By Lemma 9.1.1, H is an R-order in KCp3 , and, by Lemma 9.1.2, J is an R-order in K CO p3 . The conditions (i)–(iv) above imply that hJ; H i  R. Moreover, disc.H  / D disc.J /. (For details of these calculations, see [UC05, 3].) Thus H and J D H  are Hopf orders by Proposition 4.4.10. t u

9 Hopf Orders in KCp 3

184

The Hopf orders constructed in Proposition 9.1.1 are called duality Hopf orders in KCp3 . For p prime, the group ring KCp3 is a K-Hopf algebra. There exists a Hopf g p 7!1

inclusion KCp2 Š Khgp i ! KCp3 , and a Hopf surjection KCp3 ! Khgi Š KCp , with KCp3 = i.Khg p iC /KCp3 Š Khgi; and thus there is a short exact sequence g p 7!1

K ! Khg p i ! KCp3 ! Khgi ! K:

(9.1) 2

At the same time, there exists a Hopf inclusion KCp Š Khg p i ! KCp3 , and a 2

Hopf surjection KCp3

g p 7!1

! Khgi Š KCp2 , with

KCp3 = i.Khg p iC /KCp3 Š Khgi; 2

and thus there is a short exact sequence 2

2

K ! Khg p i ! KCp3

g p 7!1

! Khgi ! K:

(9.2)

Proposition 9.1.2. Let A.i; j; k; u; v; w/ be a duality Hopf order in KCp3 . Then there exist short exact sequences of R-Hopf orders R ! A.i; j; u/ ! A.i; j; k; u; v; w/ ! H.k/ ! R;

(9.3)

R ! H.i / ! A.i; j; k; u; v; w/ ! A.j; k; w/ ! R:

(9.4)

Proof. One shows that A.i; j; k; u; v; w/ \ Khg p i D A.i; j; u/ and that the image of A.i; j; k; u; v; w/ is H.k/ under the map given by g p 7! 1. Thus (9.3) is a short exact sequence by 4.4 (4.10). 2 Moreover, A.i; j; k; u; v; w/\Khg p i D H.i /, and the image of A.i; j; k; u; v; w/ 2 under the map given by gp 7! 1 is A.j; k; w/, and so (9.4) is a short exact sequence by 4.4 (4.10). We leave the details to the reader as an exercise. t u If H is an arbitrary R-Hopf order in KCp3 , then the extensions (9.1) and (9.2) induce extensions of R-Hopf orders, R ! A.i; j; u/ ! H ! H.k/ ! R

(9.5)

R ! H.i / ! H ! A.j; k; w/ ! R;

(9.6)

and

9.2 Circulant Matrices and Hopf Orders in KCp3

185

for R-Hopf orders A.i; j; u/ and A.j; k; w/. The sequence (9.6) dualizes to yield the short exact sequence R ! A.k 0 ; j 0 ; w/ Q ! H  ! H.i 0 / ! R:

(9.7)

In an effort to classify all Hopf orders in KCp3 , R. Underwood has given the following analog of Definition 8.1.1 for Hopf orders in KCp3 . Definition 9.1.1. Let H be an R-Hopf order in KCp3 inducing the short exact sequences (9.5) and (9.7) as above. Let „.A.i; j; u// denote the p-adic obgv determined by A.i; j; u/, and let „.A.k 0 ; j 0 w// Q denote the p-adic obgv given by A.k 0 ; j 0 ; w/ Q (see Proposition 5.3.9). Then H satisfies the valuation condition for n D 3 if either pk  „.A.i; j; u//.g p / or p Q /: pi 0  „.A.k 0 ; j 0 ; w//.

To see Definition 9.1.1 as an extension of Definition 8.1.1, let R ! H.i / ! H ! H.j / ! R and

R ! H.j 0 / ! H  ! H.i 0 / ! R

be short exact sequences, where H is an R-Hopf order in KCp2 . Then H satisfies the valuation condition for n D 2 if and only if either pj  „.H.i //./; hi D Cp ; or

pi 0  „.H.j 0 //./; hi D CO p ;

since „.H.i //./ D i and „.H.j 0 //./ D j 0 . In [UC05], the authors show that every duality Hopf order A.i; j; k; u; v; w/ satisfies the valuation condition for n D 3 (see [UC05, Theorem 3.8]), and it has been conjectured that an arbitrary Hopf order in KCp3 satisfies the valuation condition. Indeed, there are no known examples where the condition fails.

9.2 Circulant Matrices and Hopf Orders in KCp3 In this section, we construct another collection of Hopf orders in KCp3 . We keep the notation of the previous section: g denotes the image of g under the mapping 2 KCp3 ! KCp2 , g p 7! 1, and  denotes the image of  under the mapping

9 Hopf Orders in KCp 3

186

2 K CO p3 ! K CO p2 ,  p 7! 1. For an integer n  1, let h ; in denote the duality map K CO pn  KCpn ! K. For an integer m, 0  m  e 0 , let m0 D e 0  m, and for 1 a unit u 2 R, let uQ D p1 . 2 u Let # " 2 gp  1 gp au  1 A.i; j; u/ D R ; i j



and A.j; k; w/ D R

g p  1 gaw  1 ; j k



be Hopf orders in KCp2 with linear duals 

  p  1  auQ  1 ; ; auQ 2 Kh p i; A.i; j; u/ D A.j ; i ; uQ / D R j 0 i0 # " 2  p  1  p awQ  1 2  0 0 ; awQ 2 Kh p i: A.j; k; w/ D A.k ; j ; w/ Q DR ;  k0 j 0 

0

0

We assume that j 0 > pi 0 , j C i 0 > ord.1  uQ / and j 0 C k > ord.1  w/. We construct our collection of Hopf orders by choosing generators ‰ 2 KCp3 and ˆ 2 K CO p3 such that the R-modules H D A.i; j; u/Œ‰ and J D Q are invariant under the comultiplications of KCp3 and K CO p3 , A.k 0 ; j 0 ; w/Œˆ respectively. We also require that hJ; H i3  R. Then, as we shall see, H and J D H  are Hopf orders in KCp3 . We begin with the construction of ‰. Let pmCn

p1 p1 1 X X .pmCn/.paCb/ p.paCb/ D 2 p 2  ; p aD0

m; n D 0; : : : ; p  1;

bD0

q D

p1 1 X qn pn   ; p nD0 p

0  q  p  1;

and epmCn

p1 p1 1 X X .pmCn/.paCb/ p.paCb/ D 2 p 2 g p aD0 bD0

denote the idempotents for Kh i Š K CO p2 , Kh p i Š K CO p , and Khg p i Š KCp2 , respectively. Let spmCn , m; n D 0; : : : ; p  1, be units of R with spm D uQ m , and set p

D

p1 p1 X X mD0 nD0

spmCn pmCn ;

 2 Kh p i;

9.2 Circulant Matrices and Hopf Orders in KCp3

d D

p1 X

s11 spqC1 q ;

187

d 2 Kh p i:

qD0

Let xpmCn , m; n D 0; : : : ; p  1, be indeterminate, and set xD

p1 p1 X X

xpmCn epmCn ;

xpm D wm :

mD0 nD0

We seek values for xpmCn , n > 0, such that D

2

. p  1/q . p awQ  1/r .   1/s ; gx  1

E 3

D0

(9.8)

for q; r; s D 0; : : : ; p  1. Lemma 9.2.1. The solution to (9.8) is a vector .xpmCn /, 0  m; n  p  1, of elements of R defined as n pm n xpmCn D pn d ; aw i1 : 3 s1 h

Proof. One has p1  X

2

. p 1/q . p awQ 1/r . 1/s D

pn 1

q  r pmCn p2 wQ n 1 .spmCn 1/s pmCn

m;nD0

8 < 1  t  ma if n D 1, b D t 3 h t pmCn ; gepaCb i D p p p : 0 otherwise,

and

and thus (9.8) expands to .p  1/

q

p1  X mD0

pmC1 Q p2 w

!   p1 s r X X 1 t ma s st t 1 xpaCt D 0:  3 .1/ spmC1 p p p t t D0 aD0 (9.9)

For integers l; n D 0; : : : ; p  1, let .n/

hl

D

p1 1 X lq n  s : p qD0 p pqC1

9 Hopf Orders in KCp 3

188

Then (9.9) can be rewritten as ! ! p1 r s X X X r s .n/ ry y y xplCn hly D 0; .1/ p2 wQ .1/sn pn3 y n yD0 nD0

(9.10)

lD0

.n/

where the subscripts on hly are read modulo p. Equation (9.10) is equivalent to the system   8 .n/ .n/ .n/ n ˆ x C h x C    C h x h D1  n pCn .p1/pCn 3 ˆ 0 1 p1 p ˆ ˆ ˆ ˆ ˆ   ˆ ˆ .n/ .n/ .n/ ˆ n ˆ D1  w Q 2 hp1 xn C h0 xpCn C    C hp2 x.p1/pCn ˆ 3 p p ˆ ˆ ˆ ˆ ˆ <   .n/ .n/ .n/ pn3 w Q p2 /2 .hp2 xn C hp1 xpCn C    C hp3 x.p1/pCn D 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ :: ˆ ˆ ˆ : ˆ ˆ ˆ ˆ ˆ   ˆ ˆ .n/ .n/ :  n w p1 .n/ / .h x C h x C    C h x Q D 1: 2 n pCn .p1/pCn p 1 2 0 p3

(9.11)

In matrix form, (9.11) appears as 0

.n/

h0 Bh.n/ B p1 B .n/ Bhp2 B B :: @ : .n/ h1

.n/

h1 .n/ h0 .n/ hp1 :: : .n/ h2

0 1 0 1 .n/ .n/ 1 pn xn 3 h2 : : : hp1 B C B C .n/ .n/ C xpCn C B wpn 3 h1 : : : hp2 C CB B B C C 2 n C .n/ .n/ C B B C w  C h0 : : : hp3 C B x2pCn C D B p3 C : C B C :: :: C B :: B C B C : : A@ : A @ A .n/ .n/ h3 : : : h0 wp1 pn x.p1/pCn 3

(9.12)

Here the coefficient matrix is the circulant matrix   .n/ .n/ .n/ .n/ M .n/ D circ h0 ; h1 ; h2 ; : : : ; hp1 : n 6D 0, for 0  q  p  1, with Note that the eigenvalues of M .n/ are spqC1 lq

corresponding eigenvectors .p /, 0  l  p  1. Thus M .n/ is invertible with .n/ inverse ‚.n/ D .m;l / for m; l D 0; : : : ; p  1. Consequently, the matrix equations in (9.12) have unique solutions for m; n D 0; : : : ; p  1, n > 0. These solutions are computed to be n pm n d ; aw i1 : xpmCn D pn 3 s1 h

t u

9.2 Circulant Matrices and Hopf Orders in KCp3

189

Now let upmCn D xpmCn , n > 0, and let bD

p1 p1 X X

upmCn epmCn ;

upm D wm :

mD0 nD0

Put ‰D

gb  1 ; k

and let

 H D A.i; j; u/Œ‰ D A.i; j; u/

gb  1 k



denote the R-module that is the A.i; j; u/-span of the set (

    ) gb  1 p1 gb  1 gb  1 2 ; ;:::; : 1; k k k

Lemma 9.2.2. Let H be the R-module as above. Suppose  p    1 d  1 ; (i) A.j 0 ; i 0 ; uQ / D R , j 0 i0  p (ii) ord p2 s1 h p d p ; aw i1  1  pi 0 C k, and k (iii) ord.p3 s1  1/  i 0 C 2 . p Then H is invariant under the comultiplication of KCp3 . Proof. (Sketch.) We show that KCp3 .H /  H ˝R H . Since KCp3 .‰/ D ‰ ˝ 1 C 1 ˝ ‰ C  k ‰ ˝ ‰ C

.gb/  gb ˝ gb ; k

H is invariant under D KCp3 if and only if .gb/  gb ˝ gb D k



.b/  b ˝ b k

 .g ˝ g/ 2 H ˝R H:

The condition j 0 > pi 0 guarantees that A.i; j; u/ 6D RCp2 , and so, by Chapter 4, Exercise 12, A.i; j; u/ is a local ring with maximal ideal .; A.i; j; u/C /. One has b 2 A.i; j; u/ and b 62 .; A.i; j; u/C /, and so b is a unit in A.i; j; u/. Thus g is a unit in H . Therefore H is invariant under KCp3 if and only if .b/  b ˝ b 2 A.i; j; u/ ˝ A.i; j; u/; k

9 Hopf Orders in KCp 3

190

which follows from the conditions of the lemma. The reader is referred to [Un08b, Lemma 2.6] for the details of this computation. t u gb  1 Our next task is to construct ˆ. Suppose that H D A.i; j; u/Œ is an k Pp1 Pp1 R-module as constructed by Lemma 9.2.2, with b D mD0 nD0 upmCn epmCn . Set Pp1 p c D mD0 u1 1 upmC1 fm , where fm are the minimal idempotents in Khg i Š KCp . The quantities c and fm are the analogs in the dual situation for d and q . Let ypmCn , m; n D 0; : : : ; p  1 be indeterminate, and set p1 p1 X X

yD

ypmCn pmCn ;

ypm D uQ m :

mD0 nD0

The ypmCn are the analogs in the dual situation for xpmCn . Then, following the construction of b as above, we find values for ypmCn , n > 0, for which 

l  2 m p p t y  1; g  1 .g au  1/ .gb  1/ D 0

(9.13)

for l; m; t D 0; : : : ; p  1, t > 0. Lemma 9.2.3. The solution to (9.13) is a vector .ypmCn /, 0  m; n  p  1, of elements of R defined as n pm n ypmCn D pn c ; auQ i 3 u1 hg

for m; n D 0; : : : ; p  1, n > 0. Proof. We follow the method of Lemma 9.2.1. For integers l; n D 0; : : : ; p  1, .n/ define l as .n/

l

D

p1 1 X lq n  u : p qD0 p pqC1

Then, finding quantities ypmCn that satisfy (9.13) is equivalent to solving the matrix equations for n D 1; 2; 3; : : : ; p  1: 0

.n/

0 B.n/ B p1 B .n/ Bp2 B B :: @ : .n/ 1

.n/

1 .n/ 0 .n/ p1 :: : .n/ 2

0 1 .n/ .n/ 1 1 pn 2 : : : p1 0 yn 3 B C .n/ .n/ B uQ pn C 1 : : : p2 C ypCn C 3 CB B B C C .n/ .n/ C B 2 n C B C C y u Q  0 : : : p3 C B 2pCn C D B p3 C : B C B C :: :: C A B ::: C A@ : : @ A .n/ .n/ y.p1/pCn uQ p1 pn 3 : : : 0 3

(9.14)

9.2 Circulant Matrices and Hopf Orders in KCp3

191

Here the coefficient matrix is the circulant matrix   .n/ .n/ .n/ .n/ N .n/ D circ 0 ; 1 ; 2 ; : : : ; p1 : .n/

Let ˆ.n/ D . m;l /, m; l D 0; : : : ; p  1 denote the inverse of N .n/ . Then the matrix equations in (9.14) have unique solutions, ypmCn D vpmCn D

pn 3

p1 X

.n/

m;l uQ l

lD0

D

n pm n pn c ; auQ i; 3 u1 hg

for m; n D 0; : : : ; p  1, n > 0.

t u

Now, let yDˇD

p1 p1 X X

vpmCn pmCn ;

vpm D uQ m ;

mD0 nD0

put ˆD and let

ˇ  1 ; i0 

ˇ  1 J D A.k ; j ; i /Œˆ D A.k ; j ; w/ Q 0 i 0

0

0

0

0



denote the R-module that is the A.k 0 ; j 0 ; w/-span Q of the set (

  )   ˇ  1 p1 ˇ  1 ˇ  1 2 : ; ;:::; 1; i0 i0 i0

Lemma 9.2.4. Suppose H satisfies the hypothesis of Lemma 9.2.2. Suppose i  pj , k 0  p 2 i 0 , j  p 2 k > pk, and e 0  i C j C k. Then the R-module J is invariant under the comultiplication of K CO p3 . Proof. We use the criteria of Lemma 9.2.2 to show that K CO 3 .J /  J ˝R J . In p

this case, J is invariant if j > pk,  g p  1 gc  1 ; A.j; k; w/ D R ; j k 

 p ord p2 u1 hg p c p ; auQ i  1  pk C i 0 ;

9 Hopf Orders in KCp 3

192

and ord.p3 u1  1/  k C

i0 ; p2

which follow from the conditions of the lemma. Details of the computations are found in [Un08b, Lemma 2.7]. h i ˇ  1 Let H D A.i; j; u/ gb1 and J D A.k 0 ; j 0 ; w/Œ Q be R-modules as k i0 constructed above by Lemma 9.2.2 and Lemma 9.2.4. We need several more lemmas before we can show that H and J are Hopf orders. Let H  D f˛ 2 K CO p3 W h˛; H i3  Rg denote the linear dual of the R-module H . Lemma 9.2.5. H  is an R-algebra. Proof. By Lemma 9.2.2, KCp3 W H ! H ˝ H . Thus there is a map of linear duals KC 3 W .H ˝R H / ! H  . Since H  ˝R H   .H ˝R H / , KC 3 serves as p p multiplication on H  . t u Lemma 9.2.6. Let H be the image of H under the mapping KCp3 ! KCp2 , 2 g p 7! 1. Then  Q K CO p2 \ H  D H D A.j; k; w/ D A.k 0 ; j 0 ; w/:   Proof. We show that H D K CO p2 \ H  . Let ˛ 2 H . Then ˛ 2 K CO p2 and h˛; H i2  R. Let f 2 H , and let f be the image of f under the mapping KCp3 ! KCp2 . Then h˛; f i3 D h˛; f i2 . Hence

h˛; H i3 D h˛; H i2  R; so that ˛ 2 H  . Hence ˛ 2 K CO p2 \ H  . Now suppose ˛ 2 K CO p2 \H  . Then ˛ 2 K CO p2 and h˛; H i3  R. Consequently,   ˛ 2 H , which shows that H D K CO p2 \ H  . t u Since b D aw , H D A.j; k; w/, which completes the proof of the lemma. Lemma 9.2.7. J  H  . Proof. By Lemma 9.2.5, H  is an algebra. Thus it suffices to show that ˇ  1 A.k 0 ; j 0 ; w/ Q  H  and 2 H  . By Lemma 9.2.6, H  \ K CO p2 D 0 i ˇ  1 A.k 0 ; j 0 ; w/, Q and thus A.k 0 ; j 0 ; w/ Q  H  . We claim that 2 H  ; but this 0 i amounts to showing that 0 hˇ  1; H i3   i R:

9.2 Circulant Matrices and Hopf Orders in KCp3

193

 auQ  1 ˇ  1 acts on A.i; j; u/ as , it suffices to show that i0 i0  

l  2 ord ˇ  1; g p  1 .g p au  1/m .gb  1/t  i 0 C li C mj C tk (9.15)

Since

for l; m; t D 0; : : : ; p  1, t > 0. But (9.15) is satisfied since (9.13) holds with y D ˇ. t u Proposition 9.2.1. Let A.i; j; u/ and A.j; k; w/ be Hopf orders in KCp2 with linear duals A.j 0 ; i 0 ; uQ / and A.k 0 ; j 0 ; w/, Q respectively. Let s be a unit of R, and put spmCn D s n uQ m for m; n D 0; : : : ; p  1. Let D

p1 p1 X X

spmCn pmCn D

mD0 nD0

Let bD

p1 p1 X X

s n uQ m pmCn :

mD0 nD0

p1 p1 X X

upmCn epmCn ;

mD0 nD0 n G.pm uQ n ; w/. where upmCn D pn 3 s Suppose p1 p1 X X ˇD vpmCn pmCn ;

vpm D uQ m ;

mD0 nD0

satisfies



l  2 m p p t ˇ  1; g  1 .g au  1/ .gb  1/ D 0

for l; m; t D 0; : : : ; p  1, t > 0: Additionally, suppose the following conditions are satisfied:  (i) ord p2 s p G.up ; w/  1  pi 0 C k; k (ii) ord.p3 s  1/  i 0 C 2 ; p (iii) i  pj ; (iv) j 0 > pi 0 ; (v) k 0  p 2 i 0 ; (vi) j  p 2 k > pk; (vii) e 0  i C j C k. i h is a Hopf order in KCp3 with linear dual Then H D A.i; j; u/ gb1 k i  h ˇ1 . J D A.k 0 ; j 0 ; w/ Q i0

9 Hopf Orders in KCp 3

194

Proof. Conditions (i)–(vii) show that KCp3 .H /  H ˝R H and K CO 3 .J /  p J ˝R J . We show that J is an R-Hopf order. By Lemmas 9.2.5 and 9.2.6, H  is an R-algebra with H  \ K CO p2 D A.k 0 ; j 0 ; w/. Q By Lemma 9.2.7, ˇ1 2 H  . Thus 0 i

satisfies a monic polynomial of degree p with coefficients in A.k 0 ; j 0 ; w/. Q Thus J is an R-algebra, and consequently J is an R-Hopf order. By Lemma 9.2.7, H  J  . Since J  \ KCp2 D A.i; j; u/, gb1 satisfies a k monic polynomial of degree p with coefficients in A.i; j; u/. Thus H is an R-Hopf order. An application of Proposition 7.1.3 then shows that H  D J . ˇ1 0 i

In light of the fact that circulant matrices play a key role in their construction, we call the Hopf orders constructed in Proposition 9.2.1 circulant matrix Hopf orders in KCp3 .

9.3 Chapter Exercises Exercises for 9.1 1. Let K be a finite extension of Qp . Show that there are Larson orders in KCp3 that are not duality Hopf orders. 2. Let K be a finite extension of Q2 . Construct an example of a duality Hopf order in KC8 . 3. In the construction of the duality Hopf orders, prove that either A.j; k; w/ or A.j 0 ; i 0 ; uQ / is a Larson order. 4. Compute the p-adic obgv determined by an arbitrary duality Hopf order. 5. Give the details in the proof of Proposition 9.1.2. Exercises for 9.2 6. 7. 8. 9. 10.

Prove that every Larson order in KCp3 is a circulant matrix Hopf order. Construct an example of a circulant matrix Hopf order in KC8 . Prove that there exist circulant matrix Hopf orders that are not duality. Prove that there exist duality Hopf orders that are not circulant matrix. Does the valuation condition for n D 3 hold for the collection of circulant matrix Hopf orders?

Chapter 10

Hopf Orders and Galois Module Theory

For this chapter, we return to the global situation where K is a finite extension of Q, R is the integral closure of Z in K, and L is a Galois extension of K with group G and ring of integers S . In this chapter, we study applications of Hopf orders to Galois module theory. Galois module theory is the branch of number theory that seeks to describe S as a module over the group ring RG. We begin with a review of some Galois theory.

10.1 Some Galois Theory Proposition 10.1.1. (The Fundamental Theorem of Algebra) Let f .x/ be a nonconstant polynomial in CŒx. Then there is a zero of f .x/ in C. Proof. Various proofs can be found (see, for example, [Rot02, Theorem 4.49]); a familiar proof that employs Liouville’s Theorem can be found in [Fr03, Theorem 31.17]. t u Proposition 10.1.2. Let p.x/ be an irreducible monic polynomial of degree m  1 in KŒx. Then the zeros of p.x/ are distinct. Proof. By Proposition 10.1.1, there exists a zero ˛ of p.x/ in C. Let ˛ W KŒx ! C be the evaluation homomorphism defined as f .x/ 7! f .˛/. Then ker.˛ / D .p.x//, and so p.x/ is the monic polynomial of smallest degree for which ˛ is a zero. By way of contradiction, assume that ˛ has multiplicity  2. Since K has characteristic 0, p 0 .x/ is a non-constant polynomial of degree m  1 < m for which ˛ is a root. This contradicts that p.x/ is a polynomial of smallest degree for which p.˛/ D 0, and thus the zeros of p.x/ are distinct. t u

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 10, © Springer Science+Business Media, LLC 2011

195

196

10 Hopf Orders and Galois Module Theory

Let p.x/ be an irreducible monic polynomial of degree m in KŒx. By repeated uses of Proposition 10.1.1 and the Factor Theorem, f .x/ factors linearly over C as p.x/ D .x  ˛1 /.x  ˛2 /    .x  ˛m /; where ˛1 ; ˛2 ; : : : ; ˛m are the distinct zeros of p.x/. Clearly, C is a field extension of K that contains all of the zeros of p.x/. There is a smallest field extension L=K that contains all of the zeros of p.x/. This is the splitting field of the irreducible polynomial p.x/. The splitting field of p.x/ is necessarily a finite extension of K; that is, the splitting field is a finite-dimensional vector space over K. Finite extensions of K are simple extensions. Proposition 10.1.3. Let L=K be a finite extension of fields. Then there exists an element  2 L for which L D K./. Proof. It suffices to prove the proposition in the case L D K.˛; ˇ/, where ˛; ˇ 2 C. Let q.x/ be the irreducible polynomial of ˛. By Proposition 10.1.2, the roots of q.x/ are distinct, and we may list them as ˛ D ˛1 ; ˛2 ; : : : ; ˛l . Let r.x/ be the irreducible polynomial of ˇ. Again, the roots of r.x/ are distinct, and we list them as ˇ D ˇ1 ; ˇ2 ; : : : ; ˇk . Since K is an infinite field, there exists an element t 2 K for which t 6D .˛i ˛/= .ˇ  ˇj / for all i; j , 1  i  l, 2  j  k. Set  D ˛ C tˇ. Then, as the reader can verify, L D K.˛; ˇ/ D K./ (see [Fr03, Theorem 51.15]). t u Isomorphisms of simple extensions of K that fix K can be characterized as follows. Proposition 10.1.4. Let L=K be a finite extension of fields. Let ˛; ˇ 2 L, and let ˇ m D ŒK.˛/ W K. The map ˛ W K.˛/ ! K.ˇ/, defined as a0 C a1 ˛ C    C am1 ˛ m1 7! a0 C a1 ˇ C    C am1 ˇm1 ;

ai 2 K;

is an isomorphism fixing K if and only if ˛ and ˇ are zeros of the same irreducible polynomial in KŒx. ˇ

Proof. Suppose ˛ W K.˛/ ! K.ˇ/ is an isomorphism that fixes K, and let p.x/ D x m C bm1 x m1 C    C b1 x C b0 be the irreducible polynomial for ˛. Then   0 D ˛ˇ .0/ D ˛ˇ .p.˛// D p ˛ˇ .˛/ D p.ˇ/; and so ˇ is a root of p.x/. Conversely, suppose p.x/ is the irreducible polynomial for both ˛ and ˇ. Let ˇ ˛ W K.˛/ ! K.ˇ/ be the map defined as a0 C a1 ˛ C    C am1 ˛ m1 7! a0 C a1 ˇ C    C am1 ˇm1 ; ˇ

Then ˛ is an isomorphism that fixes K.

ai 2 K: t u

10.1 Some Galois Theory

197

Let L=K be a finite extension of fields. The collection of all automorphisms of L is a group under function composition, denoted by Aut.L/. The group of L=K, denoted by G.L=K/, is the subgroup of Aut.L/ consisting of elements of Aut.L/ that fix K. Definition 10.1.1. A finite extension of number fields L=K is a Galois extension if L is the splitting field of an irreducible polynomial p.x/ 2 KŒx. If L=K is a Galois extension, then the group G.L=K/ is the Galois group of L/K, denoted as Gal.L=K/. Proposition 10.1.5. Let L=K be a Galois extension with Galois group G D Gal.L=K/. Then ŒL W K D jGal.L=K/j. Proof. By definition, L=K is the splitting field of an irreducible monic polynomial p.x/ 2 QŒx of degree m. Let ˇ1 be a zero of p.x/, and put E1 D K.ˇ1 /. By Proposition 10.1.4, the set of isomorphisms of E1 that fix K contributes n1 D m elements to Gal.L=K/ (in the form of elements of the permutation group Sm ). Next, let ˇ2 be a zero of p.x/ that is not in E1 (if indeed there are any such zeros). Form the extension E2 D E1 .ˇ2 /, and let n2 D ŒE2 W E1 . By Proposition 10.1.4, this contributes n2 elements of Sm to Gal.L=K/ in the form of n2 permutations that fix all of the roots of p.x/ that are in E1 and move ˇ2 to roots of p.x/ not in E1 . Next, let ˇ3 be a zero of p.x/ that is not in E2 . Let E3 D E2 .ˇ3 /, and let n3 D ŒE3 W E2 . This contributes n3 elements of Sm to Gal.L=K/. These n3 permutations must fix all of the roots of p.x/ that are in E2 and must move ˇ3 to roots of p.x/ not in E2 . We repeat this process until Ek D L for some k. Then ŒL W K D n1 n2    nk D jGal.L=K/j:

t u

We state the fundamental theorem of Galois theory. Proposition 10.1.6. Suppose L=K is a Galois extension of number fields with group G. (i) If H is a subgroup of G, then L0 D fx 2 L W h.x/ D x; 8h 2 H g is a subfield of L. Moreover, L is a Galois extension of L0 with Gal.L=L0 / Š H . If H is a normal subgroup of G, then L0 is a Galois extension of K with Gal.L0 =K/ Š G=H . (ii) Let L0 , K  L0  L be an intermediate field. Then H D fg 2 G W g.x/ D x; 8x 2 L0 g is a subgroup of G, and L is a Galois extension of L0 with group H . If L0 is a Galois extension of K, then H is a normal subgroup of G.

198

10 Hopf Orders and Galois Module Theory

Proof. We prove (i). Various proofs of (ii) can be found in [Fr03, Chapter X, 53], [Rot02, Chapter 4, 4.2], or [La84, Chapter VIII, 1]. Suppose H  G. Clearly, K  L0 . Let a; b 2 L0 , and let h 2 H . Then h.a C b/ D h.a/ C h.b/ D a C b and h.ab/ D h.a/h.b/ D ab, and so L0 is a subring of L, which implies that L0 is a subfield of L. It follows that L is a Galois extension of L0 with group Gal.L=L0 / D H . Now assume that H G G. By Proposition 10.1.3, L0 D K./ for some  2 L0 . Observe that g 1 hg 2 H for all h 2 H , g 2 G. Thus, for a 2 L0 , .g 1 hg/.a/ D a, and so h.g.a// D g.a/ for all h 2 H . Thus g.a/ 2 L0 for all g 2 G, and so g./ 2 L0 for all g 2 G. It follows that L0 is the splitting field of the irreducible polynomial of . Thus L0 =K is a Galois extension of fields. Now, let gH be an element in the factor group G=H , and let gh 2 gH . Now, for a 2 L0 , .gh/.a/ D g.h.a// D g.a/, and so the Galois action of elements in the coset gH depends only on the action of the representative g. It follows that Gal.L0 =K/ D G=H . t u Proposition 10.1.7. Let L=K be a Galois extension with group G, n D jGj. Let a 2 L=K, and let f .x/ be the irreducible Q polynomial of a over K. Then s D n= deg.f .x// is an integer with f .x/s D g2G .x  g.a//. Proof. We have K  K.a/  L. By Proposition 10.1.6(ii), there is a subgroup H of G consisting of elements that fix K.a/. Let fa D a1 ; a2 ; : : : ; al g be the zeros of f .x/. Now, with m D jH j, t D ŒG W H , Y g2G

.x  g.a// D

l Y

.x  ai /m D f .x/m D f .x/n=t

i D1

since m D n=t Since L=K is Galois, ŒL W K D n, and since L=K.a/ is Galois with group H , ŒL W K.a/ D m. Thus, by the index formula, ŒK.a/ W K D deg.f .x// D n=m. Thus n=t D m D n= deg.f .x//, and so t D deg.f .x//. t u Q Q Expanding g2G .x  g.a// in powers of x, oneQhas g2G g.a/ 2 K (why?). Thus there is a map NL=K W L P! K defined as a 7! g2G g.a/, which is the norm map of L=K. One also has g2G g.a/ 2 K, and so there is a map trL=K W L ! K P defined as a 7! g2G g.a/, which is the trace map. Note that trL=K .S /  R and NL=K .S /  R. We give some calculations of the Galois group of the splitting field of various polynomials. Let p > 2 be prime and let p.x/ D x p1 C x p2 C    C x 2 C x C 1. Then, by the Eisenstein criterion, p.x/ is irreducible over Q. Let  D p denote a primitive pth root of unity. Then the roots of p.x/ are f;  2 ;    ;  p1 g, and the splitting field of p.x/ is K D Q./. Proposition 10.1.8. With the notation above, Gal.K=Q/ Š Cp1 , where Cp1 denotes the cyclic group of order p  1.

10.1 Some Galois Theory

199

Proof. Since ŒQ./ W Q D p  1, Gal.K=Q/ is a group of order p  1. By [Fr03, Corollary 23.6], the group of units U.Fp / is cyclic, generated by the primitive element a 2 Fp . If we number the roots of p.x/ according to the rule ri D  i for i D 1; : : : ; p  1, then there is a set of relations among the roots of p.x/, a D r.p1/a : r1a D ra ; r2a D r2a ; r3a D r3a ; : : : ; rp1

(Here, the subscripts are assumed to be the least positive residue modulo p.) These relations correspond to a permutation of the subscripts 1; 2; : : : ; p  1, which can be written as   1 2  k  p1 : gD a 2a    ka    .p  1/a (Again, the subscripts in the bottom row are the least positive residue modulo p.) The powers fg i g for i D 1; : : : ; p  1 determine p  1 automorphisms of K that fix Q, and thus Gal.K=Q/ D Cp1 . t u For a less well-known example, we consider the collection of polynomials defined as follows. Let p  5 be prime, let  D p denote a primitive pth root of unity, and let a, b be complex numbers that satisfy the relations ap C b p D 1 and ab D 1:

(10.1)

Let x be indeterminate, and let 0

a B0 B B A D B0 B: @ ::

x a 0

b x a

0 b x

0 0 b

x

b

0

0

0

1 0 0C C 0C C :: C :A  a   

denote the p  p circulant matrix circ.a; x; b; 0; 0; : : : ; 0/. Then det.A/ defines a monic degree p polynomial in x, X

.p1/=2

fp .x/ D x C p

i D1

! p p  i p2i .1/ C 1; x i pi i

which factors as Y

p1

fp .x/ D

i D0

(see [FLSU08, 2]).

.x i C a C b 2i /

(10.2)

200

10 Hopf Orders and Galois Module Theory

We find specific values for a and b that satisfy the relations (10.1). Let  D p with conjugate  D 1i 3 .

p 1Ci 3 2

2

Lemma 10.1.1. p is a pth root of . Proof. Since  D cos.2=6/ C i sin.2=6/, 6 D 1. Since p  5, p 2  1 D 2 .p C 1/.p  1/  0 mod 6. Thus p 1 D 6m D 1 for some integer m, and hence 2 p D . t u Put  D p . Then ,  satisfy (10.1). Indeed, by Lemma 10.1.1,  p C   C  D 1 and   D 1, and thus the polynomial fp .x/ factors as Y

p

D

p1

fp .x/ D

.x i C  C  2i /:

i D0

Consequently, the roots of fp .x/ are r0 D   ; r1 D  1  ; r2 D  2   2 ; r3 D  3   3 ; : : : ; rp1 D  .p1/   p1 : Lemma 10.1.2. r0 D    D 1. Proof. We have p Cp D 2 cos.2p=6/. Now, since p  5, p D 6m ˙ 1 for some integer m. Thus 2 cos.2p=6/ D 2 cos.2.6m ˙ 1/=6/ D 2 cos.2 m ˙ 2=6/ D 2 cos.˙2=6/ D 1: t u By Lemma 10.1.2, fp .x/ has a rational zero r0 D 1, and so fp .x/ is not irreducible over Q. However, the polynomial fp .x/=.x C 1/ of degree p  1 is irreducible over Q, and its roots are r1 ; r2 ; : : : ; rp1 . Proposition 10.1.9. fp .x/=.x C 1/ is irreducible over Q. Proof. The reader is referred to [FLSU08, Lemma 5] for a proof.

t u

Proposition 10.1.10. Let p  5, and let K be the splitting field of fp .x/=.x C 1/ over Q. Then Gal.K=Q/ is cyclic of order p  1. Proof. We prove the proposition assuming that 2 generates U.Fp /. For a complete proof, the reader is directed to [FLSU08, Theorem 6].

10.1 Some Galois Theory

201

For 1  j  p  1, one has 2

rj2 D . j   j /2 D  2  2j C   2j C 2 D  p2j   .p2j / C 2 D  .p2j /   p2j C 2 D rp2j C 2:

(10.3)

These relations yield a permutation on p  1 letters, 

1 gD p2

2 p4

3 ::: p  6 

p1 2

1

  p  2 p 1 : p  1  4 2 pC1 2

Let W K ! K be the map defined as .r2 / D rg.2/ D r22  2:

(10.4)

Then is an automorphism of K that fixes Q. We claim that .rgi .2/ / D rgi C1 .2/ ; 8i  0: To prove this assertion, we proceed by induction on i , with (10.4) being the trivial case. Assume that .rgi 1 .2/ / D rgi .2/ . By (10.3), rgi .2/ D rg2i 1 .2/  2, and thus .rgi .2/ / D . .rgi 1 .2/ //2  2 D rg2i .2/  2 D rgi C1 .2/ ; which completes the induction proof. Now, since g has order p  1 if and only if h2i D U.Fp /, has order p  1. Thus Gal.K=Q/ D Cp1 . t u For another example, we consider the polynomial p.x/ D x p  a 2 QŒx. Proposition 10.1.11. Either the polynomial p.x/ is irreducible over Q or a is a pth power in Q; that is, there exists ˛ 2 Q for which ˛ p D a. 1

p1

Proof. It is known that the zeros of x p a consist of fa p pi gi D0 . If p.x/ is reducible 1

p1

over Q, then p.x/ D q.x/s.x/, where the zeros of s.x/ consist of fa p pi gi D1 . Thus 1 p

q.x/ is linear over Q with zero a 2 Q.

t u

Assume that a is not a pth power in Q, so that x p a is irreducible by Proposition 10.1.11. Let  D p . Since the roots of x p  a are the p elements 1

r1 D ˛; r2 D ˛; r3 D ˛ 2 ; : : : ; rp D ˛ p1 ; ˛ D a p ; the splitting field K of x p  a is Q./.˛/ D Q.; ˛/. We compute the Galois group of K over the base field Q./.

202

10 Hopf Orders and Galois Module Theory

Proposition 10.1.12. Gal.K=Q.// D Cp . Proof. Since a is not a pth power in Q./, p.x/ D x p  a is irreducible over Q./. Thus ŒK W Q./ D p. Since K is the splitting field of x p  a, Gal.K=Q.// has order p and thus is isomorphic to Cp . The p mappings gj .˛/ D ˛ j , 0  j  p  1, determine p automorphisms of K that fix Q./, and so Gal.K=Q.// is generated by g1 . As a permutation of the subscripts of the ri , g1 appears as   1 2 3  p  1 p ; 2 3 4  p 1 t u

which is a cycle of length p.

10.2 Ramification Let K be a finite extension of Q with ring of integers R, and let p be a rational prime. By 4.2, Exercises 9 and 10, the principal ideal .p/  R factors uniquely into a product of prime ideals, e

.p/ D P1e1 P2e2    Pg g :

(10.5)

This factorization takes on a nicer form in the case where K is a Galois extension. Proposition 10.2.1. Let K=Q be a Galois extension with group G. Let p be a rational prime, and let Pi be a prime ideal in the factorization (10.5). For each h 2 G, h.Pi / D Pj for some j , 1  j  g. Moreover, for each prime ideal Pj in the factorization (10.5), there exists an element h 2 G for which h.Pi / D Pj . Proof. Let h 2 G. It is easy to show that h.Pi / is a prime ideal of R. From the factorization (10.5), we obtain e

h..p// D h.P1e1 P2e2    Pg g / D h.P1 /e1 h.P2 /e2    h.Pg /eg ; which equals .p/ since h fixes Q. Since the factorization (10.5) is unique, one concludes that h.Pi / D Pj for some j , 1  j  g. For the second statement of the proposition, let G D fh1 ; h2 ; : : : ; hn g. Suppose Pj is a prime factor of .p/ for which Pj 6D hl .Pi / for all 1  l  n. Renumbering the elements of G if necessary, let fh1 .Pi /; h2 .Pi /; : : : ; h˛ .Pi /g be the collection of distinct images of Pi under elements of G.

10.2 Ramification

203

Let l; m be integers with l 6D m, and suppose that hl .Pi / C hm .Pi /  Q for some prime Q of R. Then, hl .Pi /  Q and hm .Pi /  Q. Since R is a Dedekind domain, hl .Pi / and hm .Pi / are maximal, and so hl .Pi / D Q D hm .Pi /, which is a contradiction. Thus, by Proposition 1.1.1, hl .Pi / C hm .Pi / D R for l 6D m, 1  l; m  ˛. Put I D h1 .Pi /h2 .Pi /    h˛ .Pi /. By the Chinese Remainder Theorem for Rings [IR90, Chapter 12, Proposition 12.3.1], R=I Š R= h1 .Pi /  R= h2 .Pi /      R= h˛ .Pi /:

(10.6)

One has Pj C I D R, and so there exists an element a 2 Pj for which a  1 .mod I / and, by the Q isomorphism in (10.6), a  1 .mod hl .Pi // for 1  l  n. Thus NK=Q .a/ D nmD1 hm .a/ 2 Pj . Necessarily, NK=Q .a/ 2 Pj \ Z D pZ, and so NK=Q .a/ 2 Pi , which says that hl .a/ 2 Pi for some l. Thus a 2 hm .Pi / for some m, which is a contradiction. Thus no factor Pj exists with the property that Pj 6D hl .Pi / for all 1  l  n. t u Proposition 10.2.2. Let K=Q be a Galois extension with group G, and let p be a prime of Z. Then .p/ D .P1 P2    Pg /e for an integer e  1. Proof. Let

e

e

.p/ D P1e1 P2e2    Piei    Pj j    Pg g be the factorization of p. By Proposition 10.2.1, there exists h 2 G for which h.Pi / D Pj . Thus, .p/ D h.P1 /e1 h.P2 /e2    Pjei    h.Pj /ej    h.Pg /eg ; and so, by the uniqueness of factorization, ei D ej for all 1  i; j  g.

t u

The integer e in Proposition 10.2.2 is the ramification index of p in R. Let K be a finite Galois extension of Q with ring of integers R. By Proposition 10.1.3, we can write K D Q.˛/ for some ˛ 2 K. Let p be a prime of Z. By Proposition 10.2.2, .p/ D .P1 P2    Pg /e for some integer e  1. Let Zp denote the completion of Z with respect to j jp , with fraction field Qp . By Proposition 5.1.5, each prime Pj , 1  j  g, corresponds to a discrete absolute value Œ p;j that extends j jp . Let KPj denote the completion of K with respect to Œ pj . By Proposition 5.3.11, KPj is a finite extension of Qp . Moreover, by [FT91, Chapter III, 1, Theorem 17], there is an isomorphism Qp ˝Q K !

g M

KPj ;

(10.7)

j D1

where the primitive idempotent j of Qp ˝Q K corresponds to the prime ideal Pj .

204

10 Hopf Orders and Galois Module Theory

Proposition 10.2.3. With the notation above, g X ŒKPi W Qp :

ŒK W Q D Œ.Qp ˝Q K/ W Qp  D

i D1 m n1 Proof. Let n D ŒK W Q. Since f˛ m gn1 mD0 is a basis for K over Q, f1 ˝ ˛ gmD0 is a spanning set for Qp ˝Q K over Qp ; Qp ˝Q K is a finite-dimensional Qp -vector space. By Lemma 5.1.1,

Œ.Qp ˝Q K/ W Qp  D

l X ŒQp .˛i / W Qp : i D1

Here ˛i is a zero of the irreducible factor ri .x/ in the factorization q.x/ D r1 .x/    P rl .x/ over Qp . Thus nP D li D1 ŒQp .˛i / W Qp , and so n D Œ.Qp ˝Q K/ W Qp . Now, g by (10.7), ŒK W Q D j D1 ŒKPj W Qp . t u For i , 1  i  g, let RPi denote the completion of R at Pi . Then RPi =Pi RPi is a field extension of Zp =pZp Š Fp . Put fi D ŒRPi =Pi RPi W Fp . Let ni D ŒKPi W Qp . Proposition 10.2.4. With the notation above, ni D efi for 1  i  g. Proof. Put Ni D Pi RPi . There is an isomorphism of Fp -vector spaces RPi =pRPi Š RPi =Ni ˚ Ni =Ni2 ˚ Ni2 =Ni3 ˚    ˚ Nie1 =Nie :

(10.8)

Now, since RPi is free and of rank ni over Zp , the dimension of the vector space RPi =pRPi over Fp is ni . Now, as vector spaces over Fp , j C1

j

RPi =Ni Š Ni =Ni

for j D 1; : : : ; e  1. Thus, the right-hand side in (10.8) is the direct sum of e vector spaces, each of dimension fi over Fp , and thus the direct sum has dimension efi over Fp . Consequently, ni D efi . t u Proposition 10.2.5. Let K=Q be a Galois extension with group G, n D jGj. Let .p/ D .P1    Pg /e , and let fi D ŒRPj =Pj RPj W Fp , 1  j  g. Then f1 D f2 D    D fg and n D efg, where f D f1 . Pg Proof. By Proposition Pg10.2.3, n D i D1 ŒKPi W Qp , and, by Proposition 10.2.4, Pg i D1 ŒKPi W Qp  D i D1 efi . As in the proof of Proposition 10.2.4, j

j C1

RPi =Ni Š Ni =Ni

j

j C1

Š Nk =Nk

Š RPk =Nk

for all i; k. Thus all of the fi are equal to a common value, say f . Thus n D P g u t i D1 ef D efg.

10.2 Ramification

205

To p illustrate the usefulness of the formula n D efg, take K D Q.i/, where i D 1. We know that Q.i/ is a Galois extension of Q with group C2 . The ring of integers of K is R D ZŒi, the ring of Gaussian integers. We have already shown how prime ideals of Z factor in ZŒi (see 1.2). For example, .2/ D P12 with P1 D .1Ci/, and thus e D 2, f D g D 1. Thus (10.7) is Q2 ˝Q Q.i/ D Q.i/.1Ci/ ; and (10.8) is ZŒi.1Ci/ =2ZŒi.1Ci/ D ZŒi.1Ci/ =.1 C i/ZŒi.1Ci/ ˚ .1 C i/ZŒi.1Ci/ =.1 C i/2 ZŒi.1Ci/ : Moreover, the ideal .5/ factors as .5/ D P1 P2 with P1 D .2 C i /, P2 D .2  i /. In this case, e D 1, f D 1, and g D 2, and therefore Q5 ˝Q Q.i/ D Q.i/.2Ci/ ˚ Q.i/.2i/ D Q5 ˚ Q5 ; and ZŒi.2Ci/ =5ZŒi.2Ci/ D ZŒi.2Ci/ =.2 C i/ZŒi.2Ci/ D Z5 =5Z5 Š Z=5Z since i 2 Z5 by Hensel’s Lemma. Let K=Q be a Galois extension with group G and ring of integers R. Suppose .p/ D .P1    Pg /e , and let P be one of the prime ideals Pi . Proposition 10.2.6. KP is a Galois extension of Qp . Proof. Let K D Q.˛/, and let q.x/ be the irreducible polynomial of ˛. Let r1 .x/r2 .x/    rl .x/ be the factorization of q.x/ over Qp into irreducible polynoL mials, and let ˛i be a zero of ri .x/. Since there is an isomorphism li D1 Qp .˛i / Š Ll i D1 KPi (see Proposition 5.1.5), KP Š Qp .˛i / for some i . Necessarily, KP is the splitting field of the polynomial ri .x/. Thus, KP =Qp is a Galois extension. t u The Galois group of KP =Qp has an elegant characterization. First, a definition: the decomposition group of P is the subgroup of G defined as GP D fg 2 G W g.P / D P g: Proposition 10.2.7. GP Š Gal.KP =Qp /. Proof. Let p be the prime of Z lying below P , and let Œ p be the extension of j jp corresponding to the prime P . Let h 2 GP . The function fh W K ! K defined by fh .x/ D h.x/ is continuous with respect to the Œ p -topology on K. Thus fh extends to a function fQh W KP ! KP that is defined as   fQh lim an D lim .fh .an // D lim .h.an //; n!1

n!1

where fan g is a Πp -Cauchy sequence in K.

n!1

206

10 Hopf Orders and Galois Module Theory

Since fQh is an automorphism of KP that fixes Qp , the Galois group of KP =Qp is the group of functions ffQh W h 2 GP g, which is isomorphic to GP . t u With respect to the Galois group GP , we define the trace map trKP =Qp W KP ! Qp by the rule X trKP =Qp .a/ D h.a/ h2GP

for a 2 KP . Let f D ŒRP =PRP W Fp . The finite field extension RP =PRP over Fp has p f f elements that are precisely the p f distinct zeros of the polynomial x p x in Fp Œx. Moreover, RP =PRP is the splitting field of an irreducible polynomial q.x/ 2 Fp Œx of degree f . Thus .RP =PRP /=Fp is a Galois extension whose group is computed as follows. The group G P D fg 2 G W g.x/  x .mod P /; 8x 2 Rg is the inertia group of P . Observe that G P is a normal subgroup of GP . Proposition 10.2.8. JP D GP =G P Š Gal..RP =PRP /=Fp /. t u

Proof. Exercise.

The group JP is cyclic of order f and is generated by the automorphism  of RP =PRP , defined as .a/ D ap , where a is the image of a 2 RP under the canonical surjection RP ! RP =PRP . Note that  f .a/ D a for all a 2 RP =PRP . By Proposition 10.2.4, ŒKP W Qp  D jGP j D ef , and so, since jJP j D f , jG P j D e. Set E D RP =PRP . Since E has Galois group hi over Fp , one can define the trace function trE=.Fp / W E ! Fp as trE=.Fp / .a/ D

f X

2

 j .a/ D a C ap C ap C    C ap

f 1

:

j D1

Proposition 10.2.9. Let % W Zp ! Zp =pZp Š Fp denote the canonical surjection. Then %.trKP =Qp .a// D e  trE=.Fp / .a/

(10.9)

for a 2 RP . Proof. This follows from the decomposition (10.8) with P D Pi .

t u

Proposition 10.2.10. With the notation above, trE=.Fp / .E/ D Fp :

(10.10)

10.2 Ramification

207

Proof. Let b 2 Fp . Since trE=.Fp / .0/ D 0, we assume that b 6D 0. The polynomial 2 f 1 has at most p f 1 zeros in RP =PRP , but since E has x C xp C xp C    C xp f f 1 p >p elements, there must be an element d 2 RP for which trE=.Fp / .d / D c 6D 0. Now, trE=.Zp Z/ is surjective since trE=.Fp / .bd=c/ D .b=c/tr.E=Fp / .d / D .b=c/c D b:

t u

We next discuss the notion of relative ramification. Let K be a finite extension of Q with ring of integers R, and let L=K be a Galois extension of fields with group G, n D jGj. Let S denote the ring of integers in L. Let Q be a prime ideal of R lying above the prime p 2 Z. Analogous to Proposition 10.2.2, the ideal QS  S factors uniquely as QS D .P1 P2    Pg /eL=K ;

(10.11)

where Pi are distinct prime ideals of S and eL=K is a positive integer. We call the integer eL=K the relative ramification index of Q in S . We say that Q ramifies in L if eL=K > 1, and that Q is unramified in L if eL=K D 1. The prime Q is tamely ramified in L if p does not divide eL=K ; Q is totally ramified in L if eL=K D ŒL W K. P Recall that the trace map trL=K W L ! K is defined as trL=K .x/ D g2G g.x/, x 2 L. Analogous to Lemma 4.2.3, there exists a basis fx1 ; x2 ; : : : ; xn g for L over K that is contained in S . Let .trL=K .xi xj // denote the n  n matrix whose i; j th entry is trL=K .xi xj /. Definition 10.2.1. The discriminant of S over R is the unique ideal disc.S=R/ of R that is generated by the set fdet.trL=K .xi xj //g; where fxi g runs through all of the bases for L over K that are contained in S . Let S D D fx 2 L W trL=K .xS /  Rg denote the dual module of S . Since S D is a finitely generated R-module with S  S D , S D is a fractional ideal of L, and hence .S D /1 D fx 2 L W xS D  S g is an integral ideal of S , which we call the different. The different is denoted as D. Suppose that P is one of the primes in the decomposition (10.11). Let Œ p;P be the extension of j jp to L that corresponds to the prime P . Since p is also the unique prime lying below Q, there is an extension Œ p;Q that extends j jp to K. In fact, the restriction of Œ p;P to K is Œ p;Q , and so Œ p;P is an extension of Œ p;Q to L.

208

10 Hopf Orders and Galois Module Theory

Let LP denote the completion of L with respect to Œ p;P , and let KQ denote the completion of K with respect to Œ p;Q . Analogous to Proposition 10.2.7, Gal.LP =KQ / Š GP , where GP is the decomposition group GP D fh 2 G W h.P / D P g: The norm of LP =KQ is the map NLP =KQ W LP ! KQ defined as NLP =KQ .s/ D

Y

g.s/

g2GP

for s 2 LP , and the trace of LP =KQ is the map trLP =KQ W LP ! KQ defined as trLP =KQ .s/ D

X

g.s/:

g2GP

Let f D ŒSP =PSP W RQ =QRQ , q D ŒRQ =QRQ W Fp . The finite field extension RP =PRP over Fp has p q elements, and the extension SP =PSP over RQ =QRQ has .p q /f D p qf elements that are precisely the p qf distinct zeros of qf the polynomial x p  x in Fp Œx. Moreover, SP =PSP is the splitting field of an irreducible polynomial q.x/ 2 .RQ =QRQ /Œx of degree f with Galois group JP D Gal..SP =PSP /=.RQ =QRQ // D GP =G P ; where G P D fg 2 G W g.x/  x .mod P /; 8x 2 S g is the inertia group. The group JP is cyclic of order f and is generated by the automorphism  q of SP =PSP defined as .a/ D ap , where a is the image of a 2 SP under the canonical surjection SP ! SP =PSP . Note that  f .a/ D a for all a 2 SP =PSP . The element  lifts to an element g 2 GP for which g.a/  ap

q

mod P

for all a 2 S . Let E D SP =PSP , F D RQ =QRQ . Since E has Galois group hi over F , one can define the trace function trE=F W E ! F as trE=F .a/ D

f X

2

 j .a/ D a C ap C ap C    C ap

f 1

:

j D1

Let % W RQ ! F denote the canonical surjection. Then, analogous to (10.9), %.trLP =KQ .a// D eL=K  trE=F .a/

(10.12)

10.2 Ramification

209

for a 2 SP . Moreover, analogous to (10.10), trE=F .E/ D F:

(10.13)

Note that SP is a finitely generated torsion-free module over the PID RQ . Thus SP is free over RQ and of finite rank m D ŒLP W KQ . Let fx1 ; x2 ; : : : ; xm g be an RQ -basis for SP . The discriminant of SP =RQ is the RQ -ideal det.trLP =KQ .xi xj //RQ . Let SPD denote the dual module of SP ; that is, SPD D fx 2 LP W trLP =KQ .xSP /  RQ g: As in the global case, the different is the SP -ideal ˚  .SPD /1 D x 2 L W xSPD  SP ; which we denote as DSP . Observe that DSP D SP ˝R D D .SPD /1 . The discriminant and the different are related in the following way. Proposition 10.2.11. disc.SP =RQ / D NLP =KQ .DSP /. Proof. For a proof, see [Se79, Lemma I.6.3].

t u

A lower bound for the P -order of D can now be computed. Proposition 10.2.12. Let L be a Galois extension of K with ring of integers S ,R, respectively. Let P be a prime ideal of S that lies above the prime ideal Q in R, and let e denote the ramification index of Q in S . Then ordP .D/  e  1. Proof. We have the quotients SP =PSP , SP =QSP , and PSP =QSP D PSP =.PSP /e : For x 2 SP , let xQ denote the image of x under the canonical surjection SP ! ef SP =QSP . Let f DŒSP =PSP W RQ =QRQ . There exists an RQ =QRQ -basis fai gi D1 for SP =QSP , for which fai g, 1  i  .e  1/f , is an RQ =QRQ -basis for PSP =QSP . By Nakayama’s Lemma, there exists an RQ -basis fxi g for SP for which xi  ai mod QSP for 1  i  ef . Now, for i D 1; 2; : : : ; .e  1/f , and j D 1; 2; : : : ; ef , xi xj 2 PSP , and so trLP =KQ .xi xj / 2 QRQ . Thus, ordQ .disc.SP =RQ //  .e  1/f; and so, by Proposition 10.2.11, 1 ordQ .NLP =KQ .DSP //  .e  1/: f

210

10 Hopf Orders and Galois Module Theory

As an integral ideal of SP , DSP factors as DSP D .PSP /l , and hence ordP .DSP / D l. But since NLP =KQ .PSP / D .QRQ /f , ordQ .NLP =KQ .DSP // D lf , and so 1 ordP .DSP / D l D ordQ .NLP =KQ .DSP //  .e  1/: f Of course, ordP .DSP / D ordP .D/, which completes the proof.

t u

We can now give the two main results in this section. Proposition 10.2.13. Let L=K be a Galois extension of number fields with group G. Let P be a prime ideal of S that lies above the prime ideal Q in R, and let e D eL=K denote the (relative) ramification index of Q in S . Then the following conditions are equivalent: (i) ordP .D/ D e  1; (ii) trLP =KQ .SP / D RQ (that is, the trace map is surjective); (iii) LP is tamely ramified over KQ (that is, p does not divide e). Proof. (i) , (ii) Put r D ordQ .trLP =KQ .SP //, v D ordP .D/. By definition, SPD D fx 2 LP W trLP =KQ .xSP /  RQ g; and so SPD \ KQ D fx 2 KQ W trLP =KQ .xSP /  RQ g D fx 2 KQ W trLP =KQ .SP /x  RQ g D .trLP =KQ .SP //1 ; so that DSP \ KQ D trLP =KQ .SP /: Thus

v e

 r or v  re:

Now, if ordP .D/ D e  1, then e  1  re, so that r D 0 and trLP =KQ .SP / D RQ . Conversely, if trLP =KQ .SP / D RQ , then r D 0 and DSP D SP . It follows that QRQ SPD \ KQ , and so e > v. Now, by Proposition 10.2.12, e > v  e  1, which yields v D e  1. (ii) , (iii) Let E D SP =PSP , F D RQ =QRQ . Suppose that trLP =KQ .SP /DRQ . Then, by (10.12), F D e  trE=F .E/, and so, by (10.13), F D e  F . Consequently, .p; e/ D 1, and thus LP =KQ is tamely ramified. On the other hand, if .p; e/ D 1, then e is a unit in F; and %.trLP =KQ .SP // D trE=F .E/ D F: Thus, trLP =KQ .SP / D RQ .

t u

10.2 Ramification

211

Proposition 10.2.14. Let L=K be a Galois extension of number fields with group G. Let P be a prime ideal of S that lies above the prime ideal Q in R, and let e denote the ramification index of Q in S . Then the prime Q ramifies in S if and only if Q divides disc.S=R/. Proof. Suppose Q does not divide disc.S=R/. Then disc.SP =RQ / D RQ , and so, by Proposition 10.2.11, 0 D ordQ .NLP =KQ .DSP //, which says that 0 D ordP .D/  e  1 by Proposition 10.2.12. Thus e D 1. Conversely, suppose that e D 1, and let fxi g be an RQ -basis for SP . Then formula (10.12) yields %.trLP =KQ .xi xj // D trE=F .xi xj /: Since trE=F .xi xj / 6D 0, %.trLP =KQ .xi xj //, and consequently %.disc.SP =RQ // is not zero in F D RQ =QRQ and therefore Q does not divide disc.S=R/. t u Remark 10.2.1. If K is a Galois extension of Q, then disc.R=Z/ is always a nontrivial proper ideal of Z. Thus, at least one prime p 2 Z ramifies in K. This deep result is due to H. Minkowski [Ne99, Theorem III.2.17]. Using the preceding propositions, we can calculate the ring of integers of several Galois extensions. Proposition 10.2.15. Let K D Q.p /. Then R D ZŒp . Proof. As a Q-vector space, Q.p / D Q ˚ Q.1  p / ˚ Q.1  p /2 ˚    ˚ Q.1  p /p2 : Let P denote the principal ideal of R generated by 1  p . Let ˛ 2 R. Then ˛ l D a0 C a1 ˛ C    C al1 ˛ l1 for integers ai 2 Z and some integer l  0. Let ˛ D ˛ .mod P /. Then ˛ l D a0 C a1 ˛ C    C al1 ˛ l1 ; and thus ˛ 2 R=P is integral over Z. Since R is finitely generated and torsion-free over Z, a PID, R is free over Z of rank p  1. Let fb1 ; b2 ; : : : ; bp1 g be a Z-basis for R  Q.p /. One has R=P  Q. Thus R=P  Z since the integral closure of Z in Q is Z. Evidently, Z  R=P . Thus R=P D Z, so that P is a prime ideal of R. The ideal P p1 factors as P p1 D .p/.z/, where z 2 R with .1  p ; z/ D 1. Thus, .p/ D P p1 is the prime factorization of .p/, and so e D p  1 is the ramification index of p in R. Since .p; p  1/ D 1, KP is tamely ramified over Qp . Let ZŒp  denote the free p2 Z-module on the basis B D f1; p ; p2 ; : : : ; p g. Then ZŒp   R; the problem is to show the reverse containment.

212

10 Hopf Orders and Galois Module Theory

Relative to the basis B, one computes disc.Zp Œp =Zp / D .p e1 /. On the other hand, we have DRP D P e1 RP by Proposition 10.2.13, and so NKP =Qp .DRP / D NKP =Qp .P e1 RP / D .NKP =Qp .PRP //e1 !e1 p1   Y i D D p e1 : 1  p i D1

But now, by Proposition 10.2.11, disc.RP =Zp / D p e1 , and so RP D Zp Œp . Consequently, R D ZŒp . t u For our next example, we consider the polynomial f3 .x/ D x 3  3x C 1, which is obtained from formula (10.2) with p D 3. Clearly, f3 is irreducible over Q. Moreover, the zeros of f3 p are r1 D   , r2 D  1  , and r3   2   2 , with  3 D ,  D .1 C i 3/=2,  D 3 . These roots are related by the equations r12 D r2 C 2, r22 D r3 C 2, and r32 D r1 C 2. Thus, the splitting field of f3 is K D Q.r1 /. We have ŒK W Q D 3, so that Gal.K=Q/ D C3 . To compute a generator for the Galois group, we proceed in a manner analogous to Proposition 10.1.10. The relations among the roots yield the permutation on three letters   1 2 3 : gD 2 3 1 Let W K ! K be the map defined as .r1 / D rg.1/ D r2 D r12  2. Then is an automorphism of K fixing Q with .rgi .1/ / D rgi C1 .1/ ; 8i  0. Consequently, has order 3 since g has order 3 in S3 . Proposition 10.2.16. Let K be the splitting field of f3 .x/ D x 3  3x C 1. Then R D ZŒr1 . Proof. Let ZŒr1  denote the free Z-module on the basis f1; r1 ; r12 g. Then disc.ZŒr1 =Z/ D det.M /Z, where M is the 3  3 matrix 1 0 tr.1/ tr.r1 / tr.r12 / C B 2 3 C M DB @ tr.r1 / tr.r1 / tr.r1 /A: tr.r12 / tr.r13 / tr.r14 / Now, in view of the relations among the zeros of f3 , 0 1 0 tr.1/ tr.r1 / tr.r2 C 2/ 3 @ @ A M D D 0 tr.r1 / tr.r2 C 2/ tr.3r1  1/ tr.r2 C 2/ tr.3r1  1/ tr.r3 C 4r2 C 6/ 6 and thus disc.ZŒr1 =Z/ D det.M /Z D .81/.

1 0 6 6 3A; 3 18

10.3 Galois Extensions of Rings

213

Since ZŒr1 R, disc.R=Z/ divides disc.ZŒr1 =Z/, and so, by Proposition 10.2.14 and Remark 10.2.1, 3 is the only prime that ramifies in R. Since 3 D ŒK W Q, 3 is totally ramified in R; that is, the ramification index of 3 in R is e D 3. We have .3/ D Q 3 for some prime ideal of Q of R. Now disc.Z3 Œr1 =Z3 / D ŒRQ W Z3 Œr1 2 disc.RQ =Z3 /; where ŒRQ W Z3 Œr1  is the module index of Z3 Œr1  in RQ . Now ŒRQ W Z3 Œr1  D .3a / for some a  0. Since disc.Z3 Œr1 =Z3 / D .81/ D .34 /, 0  a  2. If a D 2, then disc.RQ =Z3 / D Z3 , which is a contradiction. If a D 1, then disc.RQ =Z3 / D .9/, which yields ord3 .disc.RQ =Z3 // D 2, and so, by Proposition 10.2.11, 2 D ord3 .disc.RQ =Z3 // D ord3 .NKQ =Q3 .DRQ //: By Proposition 12.2.12, DRQ D Q b , where b  e  1 D 2. Thus ordQ .DRQ / D b D 2; and so, by Proposition 10.2.13, KQ =Q3 is tamely ramified, which is a contradiction. Consequently, the only possibility is a D 0, and in this case disc.Z3 Œr1 =Z3 / D disc.RQ =Z3 /, which yields RQ D Z3 Œr1 . It follows that R D ZŒr1 . t u

10.3 Galois Extensions of Rings Let R be an integral domain with field of fractions K, and let S be an R-algebra that is finitely generated and projective as an R-module. We assume that S has unity 1S and that .1R / D 1S , where W R ! S is the R-algebra structure map. The collection HomR-alg .S; S / is a group under function composition. Let G be aPfinite subgroup of HomR-alg .S; S /, and let D.S; G/ denote the collection of sums g2G ag g, ag 2 S . On D.S; G/, endow an S -module structure that is exactly the S -module structure of the P Pgroup ring S G. Define a multiplication on D.S; G/ as follows. For g2G ag g, h2G bh h 2 D.S; G/, put 0 @

X g2G

1 ag g A

X h2G

! bh h D

X

ag g.bh /gh;

g;h2G

where gh is the group product in G. The resulting S -algebra D.S; G/ is the crossed product algebra of S by G. Let EndR .S / denote the collection of R-linear maps  W S ! S . Then EndR .S / is an R-module with scalar multiplication defined as .r/.t/ D r.t/, for r 2 R,

214

10 Hopf Orders and Galois Module Theory

t 2 S , and addition defined pointwise: for , 2 EndR .S /, we have . C /.t/ D .t/ C .t/. The R-module EndR .S / is an R-algebra with multiplication defined by function composition: . /.t/ D . .t//. Definition 10.3.1. The R-algebra S is a Galois extension of R with group G if the R-module map j W D.S; G/ ! EndR .S /; defined as j.

P g2G

ag g/.t/ D

P g2G

(10.14)

ag g.t/ for all t 2 S , is a bijection.

Every Galois extension of fields is a Galois extension of rings. Indeed, let L=K be a Galois extension of fields with group G. Then G D Gal.L=K/  HomK-alg .L; L/, and the map j W D.L; Gal.L=K// ! EndK .L/;  P P defined as j g2G ag g .t/ D g2G ag g.t/ for all t 2 L, is a bijection. For the next three propositions, we specialize to the situation where L=K is a Galois extension with group G, R is the ring of integers in K, and S is the ring of integers in L. Proposition 10.3.1. (S. Chase, D. Harrison, and A. Rosenberg) S is a Galois extension of R with group G if and only if for each maximal ideal M of S , and for each non-trivial g 2 G, there exists an element t 2 S for which g.t/ 6 t .mod M /. Proof. See [CHR65, Theorem 1.3].

t u

Proposition 10.3.2. S is a Galois extension of R if and only if every prime ideal Q of R is unramified in L. Proof. Suppose S is a Galois extension of R. Since S is a Dedekind domain, every non-zero prime ideal of S is maximal. Thus Proposition 10.3.1 shows that, for each prime P of S , the inertia group G P D fg 2 G W g.t/  t .mod P /; 8t 2 S g is trivial. Thus the fixed field of G P is all of L, and so, by [CF67, Chapter I, 7, Theorem 2], each prime Q of R is unramified in L. Now suppose every prime ideal Q of R is unramified in L. Then, for each prime P lying above Q, the inertia group G P is trivial by [CF67, Chapter I, 7, Theorem 2]. Thus, by Proposition 10.3.1, S is a Galois extension of R. t u Proposition 10.3.3. S is a Galois extension of R if and only if disc.S=R/ D R. Proof. Suppose disc.S=R/ D R. Then, by Proposition 10.2.14, every prime Q of R is unramified in L, and so by Proposition 10.3.2, S is a Galois extension.

10.3 Galois Extensions of Rings

215

Conversely, if S is a Galois extension, then each prime Q is unramified, and so disc.S=R/ D R. t u We revert to the general situation where R is an integral domain with field of fractions K. If S is a Galois extension of R, then the group G, and consequently the group ring RG, acts on S . The fact that RG is an R-Hopf algebra suggests that we may be able to generalize the notion of a Galois extension of rings to Hopf algebras. Let H be an R-Hopf algebra, and let S be an R-algebra that is finitely generated and projective as an R-module and is also an H -module with action denoted by h.s/ for h 2 H , s 2 S . Then S is an H -module algebra if, for all s; t 2 S , h.st/ D

X

h.1/ .s/h.2/ .t/

(10.15)

.h/

and h.1S / D H .h/1S : The fixed ring of the H -module algebra S is defined as S H D fs 2 S W h.s/ D H .h/s; 8h 2 H g: The H -module algebra S is an H -extension of R if S H D R. Let S be an H -extension of R. Then there exists a map j W S ˝R H ! EndR .S / defined as j.s ˝ h/.t/ D sh.t/ for s; t 2 S , h 2 H . Definition 10.3.2. Let S be an H -extension. Then S is a Galois H -extension of R if the map j W S ˝R H ! EndR .S / defined as j.s ˝ h/.t/ D sh.t/ is an isomorphism of R-modules. The notion of a Galois H -extension generalizes the notion of a Galois extension: if H D RG, then S is a Galois RG-extension of R if and only if S is a Galois extension of R with group G. Let S be an H -module algebra. We define a multiplication on the R-module S ˝R H as follows: for a ˝ b; c ˝ d 2 S ˝R H , X .a ˝ b/.c ˝ d / D ab.1/ .c/ ˝ b.2/ d: (10.16) .b/

Then S ˝R H together with the multiplication (10.14) is an R-algebra, which is the smash product S ]H of S by H . An element of the smash product is written as P a]b. Proposition 10.3.4. Let S be an H -extension. Then S is a Galois H -extension of R if and only if the map j W S ]H ! EndR .S / defined as j.s]h/.t/ D sh.t/ is an isomorphism of R-algebras.

216

10 Hopf Orders and Galois Module Theory

t u

Proof. Exercise.

Let S be an H -extension of R. The image of S ]1 under the map j consists of endomorphisms of the form t 7! st for s 2 S these are endomorphisms defined by “left multiplication by s” and, for this reason, we denote the image j.S ]1/ by Sleft . Our goal R is to give a characterization of Galois H -extensions in terms of Sleft . Let ƒ 2 H , t 2 S , h 2 H . Then h.ƒ.t// D .hƒ/.t/ D .H .h/ƒ/.t/ D H .h/ƒ.t/; R R and so H S  SRH . Now, since S H D R, H S  R. Observe that H acts on j.S ]H / to produce an element of EndR .S /. Indeed, for R ƒ 2 H , h 2 H , s; t 2 S , def

.ƒ  j.s]h//.t/ D ƒ.sh.t// X D ƒ.1/ .s/ƒ.2/ .h.t// by (10.13) .ƒ/

0 D@

X

1 ƒ.1/ .s/ƒ.2/ hA .t/:

.ƒ/

Thus ƒ  j.s]h/ 2 EndR .S /. With sO D j.s]1/ 2 Sleft , one has .ƒ  sO /.t/ D ƒ.st/; R which is in R since H S  R. R R Lemma 10.3.1. H j.S ]H / D H Sleft . R R Proof. Clearly,R H Sleft  H j.S ]H /, so it remains to show the reverse containment. Let ƒ 2 H , s; t 2 S , h 2 H . Then .ƒ  j.s]h//.t/ D

X

ƒ.1/ .s/ƒ.2/ .h.t//

by the module algebra property of S

.ƒ/

D

X

ƒ.1/ .s/ƒ.2/ ..h.1/ /h.2/ .t// by the counit property

.ƒ;h/

D

X

ƒ.1/ .s/. .h.1/ //ƒ.2/ .h.2/ .t//

.ƒ;h/

D

X

.ƒ;h/

ƒ.1/ .. .h.1/ //s/ƒ.2/ .h.2/ .t//

since . .h// D .h/

10.3 Galois Extensions of Rings

D

X

217

ƒ.. .h.1/ //sh.2/ .t//

.h/

D

X

. .h.1/ //ƒ.sh.2/ .t//

.h/

D

X

.ƒ .h.1/ //.sh.2/ .t//

since ƒ is an integral

.h/

D

X

.ƒ.1/ .h.1/ //.s/.ƒ.2/ .h.2/ /h.3/ /.t/

.h;ƒ/

D

X

.ƒ.1/ .h.1/ //.s/.ƒ.2/ .h.2/ /1H /.t/

.h;ƒ/

by the coinverse property X .ƒ.1/ . .h.2/ // .h.1/ //.s/ƒ.2/ .t/ D .h;ƒ/

D

X

ƒ.1/ . .h/.s//ƒ.2/ .t/

by the counit property

.ƒ/

D ƒ. .h/.s/t/

2

D .ƒ  .h/.s//.t/; t u

which is an element of ƒ  Sleft .

Proposition 10.3.5. Let H be an R-Hopf algebra that is finitely generated and projective as an R-module, and let S be an H -extension of R. Then S is a Galois H -extension of R if and only if R H

 Sleft D S  ;

S  D HomR .S; R/:

R Proof. Suppose that H Sleft D S  . There is an isomorphism ˛ W S ˝ S  ! EndR .S / defined as ˛.s ˝ f /.t/ D sf .t/. Moreover, one has the map ˇWS˝

R H

 Sleft ! j.S ]H /;

defined as ˇ.q ˝ .ƒ  sO//.t/ D q.ƒ  sO /.t/ D qƒ.st/ X ƒ.1/ .s/ƒ.2/ .t/ Dq .ƒ/

218

10 Hopf Orders and Galois Module Theory

D

X

qƒ.1/ .s/ƒ.2/ .t/

.ƒ/

1 0 X qƒ.1/ .s/]ƒ.2/ A .t/ Dj@ .ƒ/

2 j.S ]H /; R for q; t 2 S , sO D j.s]1/, ƒ 2 H . The map j W S ]H ! EndR .S /, s]h 7! , where .t/ D sh.t/, is clearly an inclusion. So, there is a diagram S˝

R

Sleft D S ˝ S 

H

#ˇ

#˛

j.S ]H /

 EndR .R/;

which commutes since ˇ.q ˝ .ƒ  sO //.t/ D q.ƒ  sO /.t/ D ˛.q ˝ .ƒ  sO //.t/: Thus j.S ]H / D EndR .S /, so that S is a Galois H -extension. For the converse, assume that S is a Galois H -extension of R. One has an isomorphism R EndR .S / Š S ˝ H  j.S ]H /; and so, by Lemma 10.3.1, there is an isomorphism EndR .S / Š S ˝  D S˝ and thus S ˝S R  flat, S D H Sleft .

R H

R H

 Sleft ;

Sleft . Now, since the R-algebra map R ! S is faithfully t u

Proposition 10.3.6. Let R be a Dedekind domain, and let H be an R-Hopf algebra that is finitely generated and projective as an R-module. Let S be an H -extension of R. Then S is a Galois H -extension of R if and only if the map Z

˝R S ! S 

'W H

defined as '.ƒ ˝ s/.t/RD ƒ.st/, for s; t 2 S , ƒ 2 map makes sense since H S  R.)

R

H,

is an isomorphism. (This

10.3 Galois Extensions of Rings

219

Proof. Let S be a Galois H -extension of R. We first show that ' is surjective. R Let ˛ 2 S P . Then, by Proposition 10.3.5, there exist elements sO 2 Sleft and ƒ 2 H for which ƒ Os D ˛. Since .ƒ  sO/.t/ D ƒ.st/ D '.ƒ ˝ s/.t/; ' is surjective. R Let P be a prime ideal of R. By Lemma 4.3.5, HP is a free RP -module of R R finite rank and, by Proposition 4.3.2, rank. HP / D 1. Thus HP ˝RP SP Š SP as RP -modules. It follows that ' is an isomorphism of R-modules. We leave the converse to the reader. t u Proposition 10.3.6 can be applied to the case S D H  . Proposition 10.3.7. H  is a Galois H -extension of R. Proof. We know that H  is a finitely generated and projective R-algebra and an H -module algebra with .H  /H D R. With H playing the role of H  , the map 

Z

' W H ˝R

! H;

˛ ˝ ƒ 7! ˛  ƒ;

H

of Proposition 4.3.2 is now the required isomorphism of 10.3.6.

t u

Let R be a Dedekind domain, let S be an H -extension of R, and let P be a prime ideal of R. Put SP D RP ˝R S . For each prime P of R, SP is RP -free; let fxi gli D1 denoteRa basis for SP over RP . Since RP is a PID, there exists a generating integral ƒ for HP . Define discHP .SP =RP / D det.ƒ.xi xj //RP : Definition 10.3.3. The discriminant of S over R with respect to H is the unique ideal discH .S=R/ of R for which discH .S=R/RP D discHP .SP =RP / for all primes P in R. Equivalently, discH .S=R/ is the ideal of R that is generated by the set fdet.ƒ.xi xj //g; where fxi g runs through Rall of the bases for L=K that are contained in S and ƒ runs through all elements of H . These bases are precisely the minimal generating sets of S over R that become bases for SP over RP . The next proposition generalizes Proposition 10.3.3 to Hopf algebras.

220

10 Hopf Orders and Galois Module Theory

Proposition 10.3.8. Let R be a Dedekind domain with field of fractions K, and let H be an R-Hopf algebra. Let S be an R-algebra that is finitely generated and projective as an R-module and is also an H -extension of R. Suppose L D K ˝R S is a Galois .K ˝R H /-extension of K. Then S is a Galois H -extension of R if and only if discH .S=R/ D R. Proof. Let P be a prime ideal of R. It suffices to show that SP is a Galois HP extension of RP if and only if discHP .SP =RP / D RP . Let ƒ be a generating integral for HP , and suppose that SP is a Galois R HP -extension of RP . Then, by Proposition 10.3.5, HP  .SP /left D SP . Let fx1 ; x2 ; : : : ; xm g be a basis for SP over RP , and let ff1 ; f2 ; : : : ; fm g be the dual basis for SP . Let yi , i D P 1; : : : ; m, be elements of SP for which ƒ  yOi D fi for 1  i  m. Write yi D m lD1 ai;l xl for ai;l 2 RP . Then ıij D fi .xj / D .ƒ  yOi /xj D ƒ.yi xj / ! m m X X ai;l xl xj D ai;l ƒ.xl xj /; Dƒ lD1

lD1

with ƒ.xl xi / 2 RP . Since each ai;l 2 RP , the m  m matrix .ƒ.xl xj // is invertible over RP . Thus, discHP .SP =RP / D det.ƒ.xl xj //RP D RP . For the converse, we suppose that discHP .SP =RP / D RP . The RP -basis fxi g for SPP is actually a KP -basis for LP D KP ˝K L. As above, one obtains yi D m lD1 ai;l xl and ıij D

m X

ai;l ƒ.xl xj /;

lD1

with ai;l 2KP . Now, since det.ƒ.xl xj // is a unit of RP , the matrix ..ai;l // D 1 .ƒ.x has entries in RP . Thus yi 2 SP for all i , and therefore l xj // R  .S / D SP . t u P left HP

10.4 Hopf-Galois Extensions of a Local Ring We return to the complete, local situation. Let p be a prime of Z, and let K be a finite extension of Qp of degree n, with p 2 K. K is endowed with the discrete valuation ord. The ring of integers R in K is a local ring with maximal ideal Q. By Proposition 5.3.12, there exists an element  2 R with ./ D Q;  is a uniformizing parameter for R. One has ord..p// D e, and since .p/ D .1  p /p1 , e 0 D e=.p  1/ is an integer. Let Cp denote the cyclic group of order p, generated by g. For each integer i , 0  i  e 0 , there exists a Larson order H.i / in KCp with linear dual isomorphic to 0 0 H.i 0 / D RΠg1 0 , where i D e  i . i

10.4 Hopf-Galois Extensions of a Local Ring

221

In what follows, we apply the results of 10.3 to construct a Galois H.i 0 /extension of R. Choose integers i and r with 0  i < e 0 and 1  r  p  1. Set w D 1 C pi Cr  . Then w cannot be a pth power in R and the polynomial x p  w 2 RŒx is 1 irreducible by Proposition 10.1.11. Set L D K.z/, with z D w p . Then L=K is a Galois extension with group Cp . Let S denote the ring of integers in L; S is a local PID with maximal ideal P D . / and is endowed with the discrete valuation ord . One has ./ D . /eL=K for some positive integer eL=K . If eL=K D 1, then L=K is unramified, and if eL=K D p, then L=K is totally ramified. Proposition 10.4.1. L D K.z/ is a totally ramified extension of K. Proof. We have ŒL W Qp  D pn, and so, by Proposition 10.2.4, pn D .e  eL=K /.f  fL=K /; where f D ŒR=./ W Fp  and fL=K D ŒS=. / W R=./. By Proposition 10.2.4, n D e  f , and so p D eL=K  fL=K : Thus either eL=K D 1 or eL=K D p. Assume that i < e 0 . We have ! ! ! p p p .1/p1 zp1 C .1/p2 zp2 C    C z .z  1/p D w  1 C 1 2 p1 D w  1 C up.z  1/;

(10.17)

where u is a unit of S , and so p  ord .z  1/  minford .w  1/; ord .p/ C ord .z  1/g: Now, if L=K is unramified and if ord .1  w/  ord .p/ C ord .z  1/; then, in view of (10.17), ord .z  1/  e 0 , with pi C r  e C e 0 ; which is a contradiction since i < e 0 and r < p. So, either L=K is totally ramified or ord .1  w/ < ord .p/ C ord .z  1/: But, in the latter case, p  ord .z  1/ D ord .w  1/, and so ord .1  w/ is an integer divisible by p, which says that L=K is totally ramified. t u

222

10 Hopf Orders and Galois Module Theory

Proposition 10.4.2. Let x D .z  1/= i . Then RŒx is a Galois H.i 0 /-extension of R. Proof. We use Proposition 10.3.8 to show that RŒx is a Galois H.i 0 /-extension of R. We first check that the conditions of Proposition 10.3.8 are satisfied: RŒx is finitely generated and projective as an R-module, RŒx is an H.i 0 /-extension of R, and L D K ˝R RŒx is a Galois KCp -extension of K. (The verification of these statements is left to the reader.) So it is a matter of checking that discH.i 0 / .RŒx=R/ D R. 0 P By Proposition 4.4.5 and Proposition 5.3.3, ƒ D  .p1/i g2Cp g is a 0

generating integral for H.i 0 /. Note that ƒ D trL=K = .p1/i .

We now compute discH.i 0 / .RŒx=R/. Note that RŒ i x D RŒz  R z1 . We i compute disc.RŒz=R/. With respect to the basis f1; z; z2 ; : : : ; zp1 g, we have, by [La84, p. 348], disc.RŒz=R/ D .1/p.p1/=2 NL=K .pzp1 /R D p p R: Now, the matrix that multiplies the basis basis f.z  1/m g of RŒz is 0 1 0 0 B0  i 0 B B0 0  2i B B: @ :: 0

0

p1

f.z  1/m = mi gmD0 of RŒx to give the

0

1

  

0 0 0 :: :



 .p1/i

C C C C; C A

and so, by a familiar property of discriminants, p p R D  p.p1/i R  disc.RŒx=R/: Thus, disc.RŒx=R/ D

pp  p.p1/i

pp  p.p1/i

R, and so

R D det.tr.x a x b //R 0

0

D  p.p1/i det.tr.x a x b /= .p1/i /R 0

D  p.p1/i det.ƒ.x a x b //R: Thus, 0

det.ƒ.x a x b //R D  pep.p1/i p.p1/i R 0

D  pep.p1/.i Ci / R 0

D  pep.p1/e R D R: t u

10.4 Hopf-Galois Extensions of a Local Ring

223

We know that RŒx is contained in the ring of integers of L. But when precisely is RŒx the ring of integers of L? Proposition 10.4.3. Let RŒx be defined as above with 0  i < e 0 . Then RŒx is the ring of integers of L D K.z/, zp D w, w D 1 C  pi Cr if and only if r D 1. Proof. Clearly, RŒx is contained in the ring of integers in L. Since i < e 0 , [Ch87, Lemma 13.1] applies to show that ord .x/ D r, and so RŒx is the ring of integers of L if and only if r D 1. t u We summarize the discussion above. For a given R-Hopf order H.i 0 / in KCp , we have found a Galois extension L=K whose ring of integers S is a Galois H.i 0 /extension. We say that H.i 0 / is “realizable as a Galois group. ” The formal definition is as follows. Definition 10.4.1. Let H be an R-Hopf order in KG. Then H is realizable as a Galois group if there exists a Galois extension L=K with group G for which the ring of integers S in L is a Galois H -extension. N. Byott [By04, Theorem 6.1] has provided the following convenient criterion for realizability. Proposition 10.4.4. (N. Byott) Let n  1 be an integer, and let K be a finite extension of Qp with ring of integers R containing pn 2 K. Let ./ be the maximal ideal R. Let H be an R-Hopf order in KG, and suppose both H and H  are local rings. Then H is realizable as a Galois group if and only if H  is a monogenic R-algebra. Proof. (Sketch.) We prove the “if” assertion. Suppose H  is monogenic of rank p n over R. Then there exists an isogeny  W F ! G of formal groups of dimension m D p n  1 with H  Š RŒŒxF =.x/. Let c 2 ./n./2 , and let S D RŒŒx=J , where J is the ideal .1 .x/  c; 2 .x/; : : : ; m .x//: Then S is a Galois H -extension of R. Write S Š RŒŒx1 =. .x1 /  c/ with .x1 / D 1 ..c; 0; 0; : : : ; 0//. Then there exists an irreducible polynomial q.x1 / for which S Š RŒŒx1 =.q.x1 // Š RŒy=q.y/ (c 2 ./n./2 guarantees the irreducibility of q by the Eisenstein criterion). Let ˛ be a zero of q.y/ in some extension field L=K. Then S Š RŒ˛ is the ring of integers of K.˛/. It follows that H is realizable as a Galois group. t u The following are some applications of Byott’s Theorem. Proposition 10.4.5. Assume that p 2 K. Let H.i / be an R-Hopf order in KCp with 0 < i < e 0 . Then H.i / is realizable as a Galois group.

224

10 Hopf Orders and Galois Module Theory

t u

Proof. Exercise.

Proposition 10.4.6. Assume that p2 2 K. Let H D A.i; j; u/ be a Greither order in KCp2 with e 0 > ord.u 1/ D i 0 C.j=p/. Then H is realizable as a Galois group. Proof. By Proposition 8.3.9, the linear dual of H is the R-Hopf order A.i; j; u/ D R



” p  1 ”auQ  1 ; : j 0 i0

Observe that e 0 > i 0 C .j=p/  0 guarantees that A.i; j; u/ and A.j 0 ; i 0 ; uQ / are local rings, and so Byott’s Theorem will apply. We claim that  ”auQ  1 D H : R i0 i h  H  , so it suffices to show that Certainly R ”auQi1 0    p    ”  1 ”auQ  1 ”auQ  1 ; D disc R : disc R i0 j 0 i0 By Proposition 8.1.1 (applied to H  ) and Proposition 4.4.13,    p ”  1 ”auQ  1 2 0 0 2 ; D . p =.p1/.j Ci / /p R: disc R j 0 i0 i  h On the other hand, disc R ”auQi1 0 D D

1 

p2 .p2 1/i 0



p2 .p2 1/i 0

1

  2 disc 1; ”auQ  1; .”auQ  1/2 ; : : : ; .”auQ  1/p 1   2 disc 1; ”auQ ; .”auQ /2 ; : : : ; .”auQ /p 1 :

Now p 2 1

X

.”auQ / D k

.pm un /k epmCn

pmCnD0

for 0  k  p 2  1, and so 0

1 ”auQ .”auQ /2 :: :

1

C B C B C B CDM B C B C B A @ 2 p 1 .”auQ /

0 B B B B B @

e0 e1 e2 :: :

ep2 1

1 C C C C; C A

10.4 Hopf-Galois Extensions of a Local Ring

225

where M is the p 2  p 2 matrix whose .pm C n C 1/st column, 0  m; n  p  1, is 0

1

1

B C B pm un C B C B C B .pm un /2 C B C B C: B .pm un /3 C B C B C :: B C : @ A 2 m n p 1 .p u / Since disc.e0 ; e1 ; : : : ; ep2 1 / D R; it suffices to compute .det.M //2 . Since M is Vandermonde, Y

det.M / D

0

0

.pm un  pm un /:

0pmCn
But

0

0

ord.pm un  pm un / D ord.p  1/ D e 0

if n D n0 , and 0

0

ord.pm un  pm un / D ord.u  1/ D i 0 C .k=p/ for all other cases. Thus,  ord.det.M // D

 2 2   p 2 .p  1/ 0 p .p  1/ p 2 .p  1/ e C .i 0 C .k=p//;  2 2 2

and so     ”auQ  1 D p 2 .p  1/e 0 C .p 2 .p 2  1/ ord disc R i0 p 2 .p  1//.i 0 C .j=p//  p 2 .p 2  1/i 0 D p 2 .p  1/i C p 2 .p  1/j #!! " 2 ” p  1 ”auQ  1 ; ; D ord disc R j 0 i0 which completes the proof.

t u

226

10 Hopf Orders and Galois Module Theory

A realizability result for Hopf orders in KCp3 follows. i h be a Hopf order in KCp3 with Proposition 10.4.7. Let H D A.i; j; u/ gb1 k h i linear dual J D A.k 0 ; j 0 ; w/ Q ”ˇ1 , as constructed in Proposition 9.2.1. If 0 i (i) ord.1  u/ D i 0 C

j p

and

(ii) ord.p3 s  1/ D i 0 C

k , p2

then H is realizable as a Galois group.

Proof. The method here is similar to the proof of Proposition 10.4.6. For details, the reader is referred to [Un08b, 3]. t u

10.5 The Normal Basis Theorem In this section, we remain in the local situation where K is a finite extension of Qp , R is its ring of integers,  is a uniformizing parameter for R, and L is a Galois extension of K with group G and ring of integers S with uniformizing parameter . Set n D ŒL W K. The integer e D eL=K denotes the ramification index of  in S ; that is, ./ D . /e . We say that the extension L=K is tame if p does not divide e, and L=K is wild if p divides e. If e D 1, then L=K is unramified, and if e D ŒL=K, then L=K is totally ramified. Example 10.5.1. Let K D Qp , L D Qp .p /. Then .p/ D .p  1/p1 , so that e D p  1. Thus L=K is tame and totally ramified. On the other hand, if L D Qp .p2 /, then e D p.p  1/, and so L=K is wild and totally ramified. By Proposition 10.1.3, L D K.˛/, where ˛ is a zero of an irreducible polynomial q.x/ 2 KŒx of degree n. The set f1; ˛; ˛2 ; : : : ; ˛ n1 g is a K-basis for L, though obviously there are many other bases for L=K. Let G D f1 D g0 ; g1 ; g2 ; : : : ; gn1 g. Can we find a basis for L=K that is of the form fw; g1 .w/; g2 .w/; : : : ; gn1 .w/g for an element w 2 L? The answer is “yes,” and this allows us to identify L with KG. Proposition 10.5.1. Let L=K be a Galois extension with group G. Then there exists an element w 2 L for which fw; g1 .w/; g2 .w/; : : : ; gn1 .w/g is a basis for L over K. Proof. Let 1 D g0 ; g1 ; : : : ; gn1 denote the elements of G. Let fXgi1 gj g be a collection of indeterminates where 0  i; j  n  1. Note that there are only n distinct indeterminates Xg0 ; Xg1 ; : : : ; Xgn1 in this collection. Let .Xgi1 gj / denote the n  n matrix whose i; j th entry is Xgi1 gj , and put f .Xg0 ; Xg1 ; : : : ; Xgn1 / D det..Xg1 gj //: i

10.5 The Normal Basis Theorem

227

Then f is not identically 0. Now define a subset B of „ L  L ƒ‚ L … as n

B D f.g0 .x/; g1 .x/; : : : ; gn1 .x// W x 2 Lg: LL Since each gi is an automorphism of L, B D L „  L  ƒ‚ …. Now, by n

[Wat79, Chapter 4, p. 29, Theorem], f cannot vanish on all of B. Consequently, there exists an element w 2 L for which f .g0 .w/; g1 .w/; : : : ; gn1 .w// D det..gi1 gj .w/// 6D 0: Next, let a0 ; a1 ; : : : ; an1 be elements of K for which g0 .w/a0 C g1 .w/a1 C    C gn1 .w/an1 D 0: Then, for each i , 0  i  n  1, gi1 g0 .w/a0 C gi1 g1 .w/a1 C    C gi1 gn1 .w/an1 D 0; and so the homogeneous system 0 1 0 C B0 C B C B C B .gi1 gj .w// B C D B:C A @ :: A @ 0 an1 0

a0 a1 :: :

1

has only the trivial solution. It follows that fgi .w/gn1 i D0 is a K-basis for L.

t u

Proposition 10.5.1 is the classical Normal Basis Theorem. For example, we 1 1 2 suppose that K D Qp .p3 / and let L D K.2 3 /. Now, E D f1; 2 3 ; 2 3 g is a K-basis for L. Moreover, L=K is a Galois extension with group C3 D f1; g; g 2 g, where 1 1 g i .2 3 / D pi 3  2 3 for i D 0; 1; 2. The Normal Basis Theorem says that there exists an element w 2 L for which fw; g.w/; g 2 .w/g is a basis for L=K. Let us find such an element. 1 2 An element in L is of the form b D a0 C a1  2 3 C a2  2 3 for a0 ; a1 ; a2 2 K. Now 1 2 1 2 2 2 1.b/ D b, g.b/ D a0 Ca1 p3 2 3 Ca2 p3 2 3 , and g .b/ D a0 Ca1 p2 3 2 3 Ca2 p3 2 3 . We want to find conditions on ai such that the set B D fb; g.b/; g2 .b/g is linearly independent. With respect to the basis E, the coordinate matrix for B is 0 1 a0 a0 a0 B C Ba1 a1 p3 a1  2 3 C ; p A @ a2 a2 p2 3 a2 p3

228

10 Hopf Orders and Galois Module Theory

which has determinant a0 .a1 a2 p2 3  a1 a2 p3 /  a0 .a1 a2 p3  a1 a2 p2 3 / C a0 .a1 a2 p2 3  a1 a2 p3 / D a0 a1 a2 .p2 3  p3 /; and so B is linearly independent if and only if a0 a1 a2 6D 0; that is, if and only if ai 6D 0 for i D 0; 1; 2. So we choose a0 D a1 D a2 D 1, and see that w D 1 2 1 C 2 3 C 2 3 is the required element. The Normal Basis Theorem connects the structure of L to the structure of the group ring KG. The extension L=K is a module over KG with module action defined as 0 1 X X ax D @ ag g A x D ag g.x/ (10.18) g2G

for x 2 L, a D as follows.

P g2G

g2G

ag g, ag 2 K, so the Normal Basis Theorem can be restated

Proposition 10.5.2. Let L=K be a Galois extension with group G. Then L Š KG as KG-modules. Now let S be the ring of integers of L and let R be the ring of integers of K. The ring extension S=R is an RG-module with the module structure given by restriction of (10.18) to S . But do we have an analog for the Normal Basis Theorem? That is, does there exist an element w 2 S for which fgi .w/g is an R-basis for S ? In other words, when are S Š RG as RG-modules? We have the following criterion, a classical result due to E. Noether. Proposition 10.5.3. (Noether’s Theorem) Suppose L=K is Galois with group G. Then S Š RG as RG-modules if and only if L=K is tame. For example, if L D Qp .p /, then Gal.L=Qp / D Cp1 and L=Qp is tame. Thus S Š Zp Cp1 as Zp Cp1 -modules. Noether’s Theorem can be restated as follows. Proposition 10.5.4. Suppose L=KRis a Galois extension R with group G. Then S Š RG as RG-modules if and only if RG S D R, where RG is the ideal of integrals of the R-Hopf order RG in KG. Proof. Suppose S Š RG as RG-modules. Then, by Noether’s Theorem, L=K is tame. Thus, by Proposition 10.2.13, the trace map tr W S ! R is surjective. PBy the definition of the trace map, this is equivalent to ƒS D R, where ƒ D g2G g. R R t Now, since RG D Rƒ, one has RG S D R. The converse is left to the reader. u If L=K is wild, then Noether’s Theorem does not apply: S 6Š RG as RGmodules. But the question remains, can we still characterize S as a Galois module?

10.5 The Normal Basis Theorem

229

The answer is “yes,” but we will have to replace RG with a bigger order in KG, namely the associated order, which is defined as AL=K D fx 2 KG W xS  S g: To handle the wild case, we assume, as in (10.4), that K contains p , so that e 0 D e=.p  1/ is an integer. Choose an integer 0  i < e 0 , and set w D 1 C  pi C1 . 1 Then L D K.z/, z D w p , is a cyclic extension with group Cp D hgi, which is wild by Proposition 10.4.1. Let S be the ring of integers in L. By Proposition 10.4.3, S D RŒx with x D 0 0  i .z  1/, and H.i 0 / D RŒ g1 0  is an R-Hopf order in KCp with i D e  i . i Proposition 10.5.5. AL=K D H.i 0 / and S Š AL=K as AL=K -modules. Proof. By Proposition 10.4.2, S is a Galois H.i 0 /-extension of R, so there exists a bijection j W S ˝R H.i 0 / ! EndR .S /; 0 /. Let be the inverse of defined as j.s ˝ h/.t/ D sh.t/, for s; t 2 S , h 2 H.iP a j , and let f 2 AL=K . Then f 2 End .S /, and

.f / D R lD1 sl ˝ hl , for sl 2 S , Pa 0 hl 2 H.i /, and so f D lD1 sl hl .t/. Since f 2 KG \ SH.i 0 / D H.i 0 /, sl 2 R, for all l, so that f 2 H.i 0 /. Thus AL=K  H.i 0 /. Clearly, H.i 0 /  AL=K , and so AL=K D H.i 0 /. Since S is a left H.i 0 /-module, S is a right H.i /-comodule with structure map ˛ W S ! S ˝R H.i / given as

˛.x/ D

p X

hi .x/ ˝ fi ;

i D1

where fhi g is a basis for H.i 0 /, with dual basis ffi g. Now, from the map j , we obtain an R-module isomorphism  W S ˝R S ! S ˝R H.i /; given as .s ˝ x/ D .s ˝ 1/˛.x/ for s; x 2 S (see [Ch00, Chapter 1, 2, (2.9) Proposition]). In fact,  is an isomorphism of H.i 0 /-modules with H.i 0 / acting on the right factors. Now, since S is a free R-module of rank p, S Š R ˚ R ˚    ˚ R, and so ƒ‚ … „ p

/ ˝ S Š .R ˚ R ˚    ˚ R/ ˝R H.i /; .R ˚ R ƒ‚ ˚˚R … R ƒ‚ … „ „ p

p

S ˚ S ˚    ˚ S Š H.i / ˚ H.i / ˚    ˚ H.i /; ƒ‚ … „ „ ƒ‚ … p

p

230

10 Hopf Orders and Galois Module Theory

as H.i 0 /-modules. An application of the Krull-Schmidt Theorem [CR81, Introduction, 6B] shows that S Š H.i / as H.i 0 /-modules. Moreover, H.i / Š H.i 0 / as H.i 0 /-modules by Proposition 4.3.4, and so S Š H.i 0 / as H.i 0 /-modules. Since 0  i < e 0 implies 0 < i 0  e 0 , RCp H.i 0 /. u t An R-basis for H.i 0 / is (

    ) g  1 p1 g1 g1 2 ; ;:::; 1; i 0 ; 0 0  i i

and so there exists an element w 2 S D RŒx for which (

    ) g.w/  w g.w/  w 2 g.w/  w p1 1; ; ;:::; i0 i0 i0

is a basis for S over R. So we have seen that even though S 6Š RG as RG-modules, S Š H as H modules, where H is an R-Hopf order in KG. A natural question arises: Is there an analog of Proposition 10.5.4 for Hopf orders? Proposition 10.5.6. Suppose L=K is a Galois extension with group G. Let H be an R-Hopf order in KG, Rand suppose that S is an H -extension. Then S Š H as H -modules if and only if H S D R.  RProof. (Sketch.) By [CH86, Theorem 5.4], S Š H as H -modules if and only if t u H S D R. Now, by Proposition 4.3.4, S Š H as H -modules.

10.6 Chapter Exercises Exercises for 10.1 1. Prove that f .x/ D x p1 C x p2 C    C x 2 C x C 1 is irreducible over Q. 2. Let f7 .x/ denote the polynomial defined by (10.2) with p D 7. Compute the zeros of f7 .x/. 3. Let fp .x/ be the polynomial defined by (10.2). Prove that the zeros of fp .x/ are real numbers in the interval Œ2; 2. 4. Compute the Galois group of the splitting field of the polynomial x 5  3 over Q. 5. Suppose that L=K is a Galois extension of degree p. Prove that K.˛/ D L for all ˛ 2 LnK. Exercises for 10.2 p 6. Let p be a prime and let K D Q. p/.

10.6 Chapter Exercises

231 p 1C p

(a) For p  1 mod 4, show that R D ZŒ 2 . p (b) For p  3 mod 4, show that R D ZŒ p. (c) For q D 6 p, q  3, suppose that Q is a prime of R that lies above q. Prove p that K. p/Q is an unramified extension of Qq . 7. Let K be a finite extension of Q. (a) Prove that at most a finite number of primes ramify in K. (b) Find a proof of Minkowski’s result; that is, prove that at least one prime ramifies in K 6D Q. Exercises for 10.3 8. Let K=Q be a Galois extension with group G and ring of integers R. Prove that RG Š RG  as RG-modules. 9. Prove Proposition 10.3.4. 10. Finish the proof of Proposition 10.3.5. 11. Prove the converse of Proposition 10.3.6. Exercises for 10.4 12. Prove Proposition 10.4.5. Exercise for 10.5 13. Prove the converse of Proposition 10.5.4.

Chapter 11

The Class Group of a Hopf Order

11.1 The Class Group of a Number Field Let K be a finite extension of Q with ring of integers R. Let I be a fractional ideal in K, and let I 1 D fx 2 K W xI  Rg: By the discussion preceding Proposition 5.3.1, I 1 is a fractional ideal of R. Certainly, I 1 I  R. If I 1 I D R, then I is invertible. Proposition 11.1.1. Let I be a fractional ideal of R. Then I is invertible. Proof. Let P be a non-zero prime ideal of R. Then, by Corollary 4.2.2, RP is a PID. Thus IRP is a principal ideal of RP , which can be written as RPa for some a 2 RP . We have I 1 RP D .IRP /1 D .RPa /1 . Observe that RPa is invertible; 1 that is, .RPa /1 D RPa satisfies .RPa /1 RPa D RP :

(11.1)

Since (11.1) holds for all non-zero P , it follows that I 1 I D R.

t u

Let F .R/ denote the collection of all fractional ideals of R. We define a binary operation on F .R/ as follows. For I; J 2 F .R/, let I  J D IJ; P where IJ is the collection of all finite sums ab with a 2 I , b 2 J . Then F .R/ together with  is an Abelian monoid with R playing the role of the identity element. In fact, as a result of Proposition 11.1, F .R/ is an Abelian group; the inverse of I is I 1 .

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 11, © Springer Science+Business Media, LLC 2011

233

234

11 The Class Group of a Hopf Order

Every principal ideal Ra is a fractional ideal with “principal” fractional inverse Ra1 . The collection of all principal fractional ideals is a subgroup of F .R/ denoted by PF.R/. The quotient group F .R/=PF.R/ is the class group of R, denoted by C.R/. For a given finite extension K=Q, the class group C.R/ is a finite Abelian group. In this section, we shall prove this important fact using topology. We begin with some preliminaries. We know that up to equivalence classes the absolute values on Q consist of fj j2 ; j j3 ; j j5 ;    ; j j1 g:

(11.2)

By Propositions 5.1.3 and 5.1.6, each absolute value on Q extends to a finite number e of absolute values on K. For a prime p 2 Z, let .p/ D P1e1 P2e2    Pg g be the factorization of .p/, and let Œ p;i be the extension of j jp corresponding to the prime Pi . Let KPi denote the completion of K with respect to the Œ p;i -topology. The ring of integers in KPi is RO Pi . If i is a uniformizing parameter for RO Pi , then Œiei p;i D Œpp;i D jpjp D

1 ; p

and so Œi p;i D

1 : p 1=ei

We want to normalize Œ p;i so that the absolute value of i is p1fi , where fi D ŒRPi =Pi RPi W Fp . To do this, we define an absolute value on K by the rule j jp;i D Œ np;ii ;

ni D eifi ;

where ni is the local degree ŒKPi W Qp . Then j jp;i is equivalent to Œ p;i with ji jp;i D p1fi ; j jp;i is the normalized discrete absolute value on K corresponding to the prime Pi . To normalize the extensions of the ordinary absolute value j j1 , we also employ the local degree. But in this case the completion of K with respect to an extension k k1;i of j j1 is either R or C. Since the completion of Q with respect to j j1 is R, the local degree is either 1 or 2. Therefore the normalized Archimedean absolute value on K is defined for b 2 K as  kbk1;i D ji .b/j if i is a real embedding jbj1;i D kbk21;i D ji .b/j2 if i is a complex embedding; where j j is the ordinary absolute value on C.

11.1 The Class Group of a Number Field

235

Let fj j g2S be the set of normalized absolute values on K, and let fj j g2D denote the subset of normalized discrete absolute values on K. Definition 11.1.1. The ideal group IK is the free Abelian group on the collection fj j g2D of normalized discrete absolute values on K. Identifying j j with , IK consists of all sums of the form X n ; 2D

where n D 0 for all but a finite number of . The following is a critical proposition that describes fractional ideals in K as elements of IK . Proposition 11.1.2. (i) There is a 1-1 correspondence between fractional ideals in K and elements of IK . (ii) There is a 1-1 correspondence between ideals of R and elements of IK for which n  0.X (iii) The element n  2 IK corresponds to a principal fractional ideal in K if 2D

and only if there exists an element a 2 K  for which n D ordP .a/ for all discrete . Proof. We prove (i) and leave (ii) and (iii) as exercises. By definition, a fractional ideal has the form cJ for c 2 K  and some ideal J in R. Without loss of generality, we may assume that c 1 2 R. Let J D Q1n1 Q2n2    Qlnl be the unique factorization of J into prime ideals of R, and let .c 1 / D P1m1 P2m2    Pkmk be the unique factorization of the principal ideal .c 1 / into prime ideals. For i D 1; : : : l, let i be the normalized absolute value corresponding to the prime Qi , and for j D 1; : : : ; k, let !j be that for the prime Pj . Define a map ‰ W F .R/ ! IK by the rule ‰.cJ/ D

k X

.mj /!j C

j D1

Next, let

X

l X

ni i :

i D1

n  be an element of IK . Let 1 ; 2 ; : : : ; l denote the indices

2D

corresponding to integers with n < 0, and let lC1 ; lC2 ; : : : ; m be the indices with n > 0. For i D 1; : : : ; m, let Pi be the prime ideal corresponding to i . Then J D

ml Y

n

lCj PlCj

j D1

is an ideal of R. Moreover, M D

l Y i D1

Pni i

236

11 The Class Group of a Hopf Order

is an R-submodule of K that is finitely generated. By Proposition 4.2.2, M is a fractional ideal of R, which necessarily has the form cR for some c 2 K , c 1 2 R. Define a map ˆ W IK ! F .R/ by the rule ˆ

X

! n 

D cJ:

2D

One has ‰ˆ D idIK and ˆ‰ D idF .R/ , and so ‰ is a bijection.

t u

Again, we let fj j g2S be the set of normalized absolute values on K. For each  2 S , let K denote the completion of K with respect to the j j -topology on K. Let R denote the ring of integers in K . For each discrete j j , R is a compact subset of K . Moreover, R is open since K nR is a closed subset of K . Definition 11.1.2. The adele ring VK of K is the restricted product of the K ,  2 S , with respect to the family fR W  2 Dg, together with the restricted product Q topology on 2S K with respect to the family fR W  2 Dg. Q Consequently, VK consists of those vectors fa g 2 2S K for which a 2 R for all but finitely many . Q A basis for the topology on VK consists of open sets of the form 2S U , where U is open in K for all  and U D R for all but finitely many . The ring structure of VK is given componentwise. For fa g; fb g 2 VK , fa g C fb g D f.a C b / g;

fa gfb g D f.a b / g:

In fact, VK is a commutative ring. Proposition 11.1.3. There is an injection of rings K ! VK . Proof. Let a 2 K, and write a D b=c, where b; c 2 R. Since there are only a finite number of prime divisors of c, 1c 2 R for all but a finite number of . Thus fa g, with a D a for all  2 S , is an adele of K. Moreover, the map a ! fa g is a ring injection. t u Let f1 g be defined as 1 D 1 for all  2 S . Then f1 g is in VK , and so VK is a commutative ring with unity. Definition 11.1.3. The multiplicative group of units of VK is the idele group of K, denoted by JK . Proposition 11.1.4. The map C W JK ! R>0 defined as C.fa g/ D

Y 2S

is a homomorphism of multiplicative groups.

ja j

11.1 The Class Group of a Number Field

237

t u

Proof. Exercise.

The map C is called the content of JK . The kernel of the content map is denoted by JK1 . Let fa g 2 VK , and let Wfa g be the subset of VK defined as Wfa g D ffx g 2 VK W jx j  ja j for all  2 S g: Then Wfa g is a compact subset of VK . An important lemma follows. Lemma 11.1.1. For each finite extension K=Q, there exists a positive integer  such that if fa g is an idele in JK with C.fa g/ > , then Wfa g contains an idele f g in JK for which  D  for some  2 K for all  2 S . t u

Proof. For a proof, see [CF67, p. 66, Lemma].

Our immediate goal is to show that an embedding of rings K ! VK restricts to an embedding of groups K  ! JK1 . We need a lemma. For each absolute value j ju on Q listed in (11.2), let Qu denote the completion of Q with respect to j ju , and for  2 S let ju indicate that the absolute value j j on K is an extension of j ju . Let n denote the local degree ŒK W Qu , defined whenever ju. We define a norm on K=Q as follows. We have K D Q.˛/ for some ˛ 2 K with irreducible polynomial f .x/. Let 1 ; : : : ; l denote the set of embeddings of K into the splitting field Qof f .x/ over Q. We define the norm of K=Q to be the map NK=Q W K ! Q, x 7! li D1 i .x/. The following lemma relates the normalized valuations j j to the global norm. Q Lemma 11.1.2. Let b 2 K  . Then jNK=Q .b/ju D ju jb j . t u

Proof. See [CF67, Theorem, p. 59] for a proof. 

Proposition 11.1.5. The injection of K into VK restricts to an injection of K into JK1 . In other words, for b 2 K  , the idele fbg is in JK with C.fbg/ D 1. Proof. Since b 6D 0, fbg 2 JK . We have Y

0 1 Y Y @ jbj A jbj D u

2S

D

Y

ju

jNK=Q .b/ju

by Lemma 11.1.2.

u

Observe that r D NK=Q .b/ 2 Q . We claim that C.frg/ D 1, where C W JQ ! R>0 is the content map of the idele group of Q. We have rD

em ˙p1e1 p2e2    pm f

f

f

q1 1 q2 2    ql l

;

238

11 The Class Group of a Hopf Order

where all of the primes pi , qj are distinct and the ei , fj are positive integers. For discrete u not corresponding to one of the pi or qj , we have jrju D 1; for discrete u corresponding to a prime pi , we have jrju D p1ei ; and for discrete u f

i

corresponding to a prime qj , we obtain jrju D qj j . Also, jrju D r for u D 1, and so m l Y 1 Y fj C.frg/ D qj jrj1 D 1; p ei i D1 i j D1 t u

which completes the proof of the proposition. 

JK1 .

Since In view of Proposition 11.1.5, we identify K with a subgroup of JK1  VK , we can endow JK1 with the VK -subspace topology. Through the canonical surjection s W JK1 ! JK1 =K  , we give JK1 =K  in the quotient topology. Thus a set U is open in JK1 =K  if s 1 .U / is open in JK1 . Proposition 11.1.6. JK1 =K  in the quotient topology is compact. Proof. Let fa g be an adele in VK for which C.fa g/ >  ( as in Lemma 11.1.1), and let Wfa g be the compact set defined as Wfa g D ffx g 2 VK W jx j  ja j for all  2 S g: Now, let fb gK  be an element of JK1 =K . Then C.fb1 a g/ D C.fb1 g/C.fa g/ D C.fa g/ > ; and so, by Lemma 11.1.1, there exists an idele fg 2 K   JK1 with fg 2 Wfb1 a g . Thus, for all , jj  jb1 a j D jb1 j ja j ; and so jb j jj  ja j . Thus fgfb g 2 Wfa g \ JK1 . Since s.fgfb g/ D s.fb g/, the canonical surjection s restricts to a surjection s W Wfa g \ JK1 ! JK1 =K ; which is a continuous map of topological spaces. Now, by [CF67, p. 69, Lemma], JK1 is closed in VK , and therefore Wfa g \ JK1 is closed in the compact set Wfa g . Hence Wfa g \ JK1 is compact by Proposition 2.2.4, and so, by Proposition 2.2.5, s.Wfa g \ JK1 / D JK1 =K  is compact. t u Recall that IK is the ideal group of K. We endow IK with the discrete topology: an open set is any subset of IK . A basis for this topology consists of all singleton subsets of IK . Proposition 11.1.7. There is a surjective group homomorphism % W JK1 ! IK defined as X ordP .a /; %.fav g/ D 2D

11.1 The Class Group of a Number Field

239

where r D ordP .a / satisfies .a /  Pr , .a / 6 PrC1 . Moreover, the map % is a continuous map of topological spaces. Proof. For each  2X D, let  be a uniformizing parameter for R such that ord . / D 1. Let n  be an element of IK . Let a D n for  2 D, 2D

and for  2 XS nD choose av such that C.fa g/ D 1. Then fa g is an idele with n , and so % is surjective. %.fav g/ D 2D

We leave the proof that % is a continuous group homomorphism to the reader. u t Proposition 11.1.8. The class group C.R/ is isomorphic to IK =%.K /. Proof. By Proposition 11.1.2 (i), IK corresponds to the collection of fractional ideals F .R/, and, by Proposition 11.1.2 (iii), %.K  / corresponds to the subgroup PF.R/ of principal fractional ideals F .R/. t u Next, through the canonical surjection IK ! IK =%.K  /, we endow IK =%.K  / with the quotient topology. The quotient topology on IK =%.K / is equivalent to the discrete topology since IK has the discrete topology. We can now prove that the class group is finite. Proposition 11.1.9. Let K be an algebraic extension of Q. Then the class group C.R/ is finite. Proof. The surjection % W JK1 ! IK of Proposition 11.1.7 induces a continuous surjective map JK1 =K  ! IK =%.K /. Now, by Proposition 11.1.6, JK1 =K  is compact, and so, by Proposition 2.2.5, IK =%.K / is compact in the discrete topology. The collection fxg, x 2 IK =%.K  /, is an open covering of IK =%.K  /, from which we can extract a finite subcover since IK =%.K / is compact. Therefore, C.R/ Š IK =%.K  / is finite. t u The order of the finite group C.R/ is the class number of K, denoted by h.K/. If K C is the maximal real subfield of K with ring of integers RC , then the order of C l.RC / is denoted by hC .K/. Let K D Q.p /. The prime p satisfies Vandiver’s Conjecture if p does not divide hC .Q.p //. It is known that Vandiver’s Conjecture holds for primes < 4;000;000 [Wa97]. The order of C.R/ measures how far R is from being a PID. It is not hard to show that R is a PID if and only if C.R/ D 1. (Prove this as an exercise.) Said differently, C.R/ is non-trivial if and only if there exists a finitely generated R-submodule of K that is not a free R-module. In fact, since every fractional ideal in R is a projective R-module, the existence of a non-trivial element in C.R/ indicates the existence of a locally free R-submodule of K that is not free over R. Therep are many number fields p K for which C.R/ is non-trivial. For example, take K D Q. 5/. Then R D ZŒ 5. It is well-known that R is not a PID, and thus there exists a locally free R-submodule of K that is not free over R. (Can we come up with a specific example?)

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11 The Class Group of a Hopf Order

11.2 The Class Group of a Hopf Order Let K be a finite extension of Q with ring of integers R. The main result in the previous section was to prove that the class group C.R/ is finite. In this section, we generalize C.R/ to Hopf orders in KG, where G D Cpn , for a fixed integer n  0. We follow the method of A. Fr¨ohlich given in [Fro83, Chapter I, 2]. We know that R can be identified with the R-Hopf order RG in KG in the case n D 0, and so C.R/ is the class group of the R-Hopf order R1 in K1. We want to define the class group for R-Hopf orders in KG, where G is non-trivial. We have seen that C.R/ Š IK =%.K  /, where IK is the ideal group and %.K  / is the image of K  under the surjective map % W JK1 ! IK . In fact, there is a surjective map ' W JK ! IK . Let UK be the kernel of ' such that IK D JK =UK . Since %.K  / Š .K  UK /=UK , C.R/ Š IK =%.K  / Š .JK =UK /=..K  UK /=UK / Š JK =.K UK /:

(11.3)

In this form, C.R/ can be generalized to Hopf orders in KG. Let n be an integer, and let K be a finite extension of Q. Let S be the set of normalized absolute values on K, and let D be the subset of normalized discrete absolute values on K. Let K denote the completion of K with respect to j j . For discrete,  let R be the ring of integers in K . We adopt the convention that for  Archimedean we define R D K . Let H be an R-Hopf order in KG with G finite and Abelian. For  2 S , put H D R ˝R H . For our purposes (and for simplicity), we assume that K contains pn . By [Se77, Theorem 7], there are p n irreducible characters of G; they are precisely the homomorphisms ”m W G ! K; defined as ”m .g l / D pmln for 0  m; l  p n  1. By linearity, each character ”m determines a map (also denoted by ”m ) ”m W K  G ! K  ; which restricts to a homomorphism ”m W U.K G/ ! K ; where U.K G/ denotes the group of units in K G. The group of virtual characters of G, denoted by CG , is the free Abelian group on the set of irreducible p n 1 characters f”m gmD0 . Let Hom.CG ; JK / denote the collection of group homomorphisms from CG to JK . Then Hom.CG ; JK / is a group with group operation

11.2 The Class Group of a Hopf Order

241

defined pointwise: for f; g 2 Hom.CG ; JK /, .fg/.x/ D f .x/g.x/. Observe that Hom.CG ; K  / is a subgroup of Hom.CG ; JK /. Let M be an H -module. Then M is locally free and of rank one if M D R ˝R M is a free H -module of rank one for all  2 S and MK D K ˝R M is a free KG-module of rank one. In what follows, we show that M determines an element of Hom.CG ; JK /. Since M is locally free and of rank one, for each  2 S there exists an element x in M for which M D H x ; and there exists an element x0 in MK for which MK D KGx0 : Consequently, for each  2 S there is an element ;M 2 U.K G/ for which ;M x0 D x : One has ;M 2 U.H / for all but a finite number of , and so f;M g2S determines an idele in the idele group ( JKG D f g 2

Y

) U.K G/ W  2 U.H / almost everywhere :

2S

Let  D f g 2 JKG . For each  2 U.K G/, one has the map h W CG ! K ; defined as

0 h @

pn 1

X

1 nm ”m A D

mD0

p n 1

Y

.”m . //nm :

mD0

One easily checks that h 2 Hom.CG ; K / with h 2 Hom.CG ; R / for all but a finite number of . Passing to the product over all normalized absolute values, we obtain an element of Hom.CG ; JK / that we denote as h./. If  2 JKG arises from the locally free H -module M (that is, if  D f;M g2S for some M ), we write h.M /. Let Y UH D U.H /; 2S

and let y D fy g 2 UH . For each  2 S , y 2 U.K G/, and so the construction above applies to yield an element hy 2 Hom.CG ; U.R //  Hom.CG ; K /. Passing to the product yields an element h.y/ 2 Hom.CG ; UK /  Hom.CG ; JK /, where UK is the kernel of ' W JK ! IK . Evidently, h.UH / is a subgroup of Hom.CG ; JK /.

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11 The Class Group of a Hopf Order

We can now give a definition of the class group of H . Definition 11.2.1. Let n  1 be an integer, and assume that pn 2 K. Let H be an R-Hopf order in KCpn . Then, the class group of H is defined as C.H / D

Hom.CG ; JK / : Hom.CG ; K /h.UH /

Indeed, recalling the isomorphism (11.3), we see that C.H / D C.R/ in the case H D R, with Hom.CG ; JK / playing the role of JK , h.UH / playing the role of UK , and Hom.CG ; K  / analogous to K  . It is helpful to construct an example in which C.H / is non-trivial. Let K D Q.p2 / with ring of integers R. Put P D .1  p2 /. Then pR D P p.p1/ , and so e 0 hD p.i Let G D Cp D hgi, and let i be an integer 0  i < p. Then H.i / D g1 Pi

is an R-Hopf order in KG. Let  be the normalized discrete absolute value h i corresponding to P . Let H.i / D R ˝R H.i /. Then H.i / D R g1 , where i   is a uniformizing parameter for R . Let u be an element of R for which i 0 > ord.1  u/  0. Then, by Lemma 7.1.6,

R

au D

p1 X

 um em 2 R

mD0

 g1 nH.i / ; 0 e

where fem g is the set of minimal idempotents in K G. Let   D

1 au

if  does not correspond to P if  corresponds to P .

Then  D f g 2 JKG . Define M D H.i / D KG \

\

! H.i / v

2S

(see [CR81, 31B, p. 652]). Then h.M / is a non-trivial element in C.H.i //. A critical observation that we will need presently is the following. Let fŒM g be the collection of all isomorphism classes of locally free rank one H -modules M . Let M and N be representatives of classes in fŒM g. Let h.M / and h.N / be the elements of Hom.CG ; JK / determined by M and N , respectively. Lemma 11.2.1. If M Š N , then h.M / and h.N / lie in the same class in C.H /. Proof. For  2 S , there exist elements x 2 M and y 2 N for which M D H x

and N D H y ;

11.2 The Class Group of a Hopf Order

243

and elements x0 2 MK and y0 2 NK for which MK D KGx0

and NK D KGy0 :

Also, there exist unique elements ;M ; ;N 2 U.K G/, for which ;M x0 D x

and ;N y0 D y :

Now, since M Š N and MK Š NK , there exist elements ˇ 2 U.Hv / and 0 2 U.KG/, for which ˇ x D y and 0 x0 D y0 . Thus .ˇ1 ;N 0 /x0 D x ; and so ;M D ˇ1 ;N 0 D ;N 0 ˇ1 . Passing to the product over all  2 S yields h.M / D h.N / h.y/; where y 2 UH and 2 Hom.CG ; K  /.

t u

Like C.R/, C.H / is finite and Abelian. Proposition 11.2.1. Let n  1 be an integer, let K be a finite extension of Q containing pn , and let G be the cyclic group of order p n . Let H be an R-Hopf order in KG. Then the class group C.H / is a finite Abelian group. Proof. Let fŒM g denote the collection of isomorphism classes of locally free rank one H -modules M that are finitely generated and projective R-modules satisfying KM Š KG. By the Jordan-Zassenhaus Theorem [CR81, (24.1)], jfŒM gj < 1. Let ŒM ; ŒN  2 fŒM g. If ŒM  D ŒN , then h.M / 2 h.N /Hom.CG ; K  /h.UH / by Lemma 11.2.1. Thus, there is a well-defined map  W fŒM g ! C.H /; where .ŒM / is the class of h.M / in C.H /. We show that  is surjective. Let f Hom.CG ; K  /h.UH / 2 C.H /. Then f 2 Hom.CG ; JK / is determined by its values on the characters ”m , m D 0; : : : p n  1. For each m, f .”m / D f;m g is an idele in JK . There is a subset of ideles pn 1 ff;mggmD0  JK . For each  2 S , let  D ;0 e0 C ;1e1 C    C ;pn 1 epn 1 ;

244

11 The Class Group of a Hopf Order

where em are the idempotents in KCpn . Then  D f g is an idele in JKG . Define M D H  D KG \

\

! H v :

2S

Then M is an H -module that is finitely generated and projective as an R-module. Thus, M is a locally free rank one H -module. We have h.M / D f , and so .ŒM / D h.M /Hom.CG ; K /h.UH / D f Hom.CG ; K  /h.UH /:

t u

We consider the class group C.H / with H D RG, where G D Cpn . Suppose L=K is a Galois extension with group G, and assume that L=K is tame; that is, assume that each prime Q of R is tamely ramified in S , the ring of integers in L. By Noether’s Theorem, S is a locally free rank one RG-module. As such, S gives rise to a class .S / equal to the class of h.S / in the locally free class group C.RG/. Definition 11.2.2. The set of realizable classes R.RG/ in C.RG/ is the collection of classes in C.RG/ of the form .S /, where S is the ring of integers in a tame extension L=K with group G. In fact, R.RG/ is a subgroup of C.RG/. For a proof of this fact, see L. McCulloh’s paper [Mc87]. If R.RG/ is non-trivial, then there exists a Galois extension L=K with group G for which S is not free over RG. We want to find an analog of R.RG/ for Hopf orders H in KG. To do this, we employ the semilocalization of R at p, which is defined as Rp D fr=s W r; s 2 R; .s; p/ D 1g: In Rp , there are only a finite number of maximal ideals, namely those prime ideals of R that lie above p. Let Hp D Rp ˝R H; and let X .p/ be a Galois Hp -extension of Rp . Then X D K ˝Rp X .p/ is a Galois KG-extension of K. Let RX denote the integral closure of R in X . There is an R-order of the form ‚ D X .p/ \ RX ; which is a semilocal Galois H -extension of R. The collection of all semilocal Galois H -extensions of R is denoted by S GE.H; R/. By Proposition 10.3.7, H  is a semilocal Galois H -extension of R. Proposition 11.2.2. Each ‚ 2 S GE.H; R/ is a locally free rank one H -module.

11.2 The Class Group of a Hopf Order

245

Proof. R (Sketch.) One first shows that ‚ is a Galois H -extension. It then follows that H ‚ D R. Now, by [CH86, Theorem 5.4], ‚ is locally free and of rank one over H . t u By Proposition 11.2.2, ‚ corresponds to a class .‚/ 2 C.H /. There exists a map ‚ 7! .‚/.H  /1 ;

‰ W S GE.H; R/ ! C.H /; which is the class-invariant map. We now have an analog of R.RG/.

Definition 11.2.3. The image of the class-invariant map ‰.S GE.H; R// is the set of realizable classes R.H / in C.H /. By results of N. Byott [By95, Theorem 5.2] and L. McCulloh [Mc83], the image of the class-invariant map ‰ W S GE.RG; R/ ! C.RG/ is precisely R.RG/, and thus Definition 11.2.3 is a proper generalization of R.RG/. For the remainder of this section, we assume that G D Cp and K D Q.p2 /. Let R be the ring of integers in K. Then P D .1  p2 / is the unique prime ideal of R lying above p. One has ordP .p/ D e D p.p  1/, and so e 0 D p. For each j , 0 < j  p, there is an R-Hopf order H in KCp of the form  g1 : H D H.j / D R Pj 

The Hopf order H.j / is a free R-module of rank p with basis (

  2 p1 ) g1 g1 g1 ; ;:::; : 1; .1  p2 /j .1  p2 /j .1  p2 /j

The linear dual of H.j / is a rank p Hopf order of the form 

 ”1 H.j / D R ; 0 .1  p2 /j 0

h”i D CO p :

Proposition 11.2.3. There exists a semilocal Galois H.j /-extension that is integrally closed. 0

Proof. Let w D 1 C .1  p2 /pj C1 , and put L D K.z/, where w D zp . Let x D .z  1/=.1  p2 /j . Then Rp Œx is a Galois H.j /p -extension of Rp (Proposition 10.4.2). Moreover, Rp Œx is the integral closure of Rp in L (Proposition 10.4.3). One has L D K ˝Rp Rp Œx. Let S be the ring of integers in L. Then S D Rp Œx \ S , so that S is a semilocal H.j /-extension that is integrally closed. t u Proposition 11.2.4. Let K D Q.p2 / with ring of integers R. Put P D .1  p2 /. i h be an R-Hopf order in KCp . For an integer 0 < j  e 0 D p, let H.j / D R g1 Pj

246

11 The Class Group of a Hopf Order

Then every class in R.H.j // is the class of the ring of integers in a field extension L=K with group Cp . Proof. Note that H.j / D H.j 0 / 2 S GE.H.j /; R/. By Proposition 11.2.2, 0 0 C.H.j //. By Proposition 4.4.5, R to a class .H.j 1//Pin RH.j / gives rise p1 m 0 D . / , where

D H.j / H.j 0 / 0 0 mD0 ” , and, by Proposition 5.3.3, H.j 0 / p R 0 p.p1/.p1/j H.j 0 / . H.j 0 / / D Rr, where r D .1p2 / . Thus ƒ D r 0 is a generating integral for H.j 0 /. By Proposition 4.3.2, H.j 0 / Š H.j / ˝ Rƒ, and so H.j 0 / is a free rank one H.j /-module, which consequently corresponds to the trivial class in C.H.j //. Thus, the class-invariant map ‰ W S GE.H.j /; R/ ! C.H.j // reduces to ‚ 7! .‚/, and so the image of the class-invariant map R.H.j // consists of the classes .M / where M 2 S GE.H.j /; R/. By Proposition 11.2.3, there exists a class .S / 2 R.H.j // for which S is integrally closed. Let .M / be a class in R.H.j //. Now, by [By95, Theorem 5.6], there exists an element X 2 S GE.H.j /; R/ such that .X / D .M / and for which X is the full ring of integers of some Galois extension of K with group Cp . t u

11.3 The Hopf-Swan Subgroup Let K be a finite extension of Q containing pn with ring of integers R. Let S D fg denote the collection-normalized absolute values on K, and let D  S denote the subset of discrete normalized absolute values. Let H be an R-Hopf order in KG, G D Cpn , with class group C.H /. In this section, we construct an important /. Let r be an R  subgroup of C.HQ element of R that is relatively prime to H . In other words, if li D1 Piei is the R  Q kj factorization of .r/ and m j D1 Qj is that for H H , then Pi 6D Qj for all i; j . Definition 11.3.1. The Hopf-Swan module is the H -module defined as R ˝ R ˛ r; H D rH C H , R  where r 2 R is relatively prime to H H . ˝ R ˛ Proposition 11.3.1. The Hopf-Swan module r; H is a locally free rank one H -module. Proof. For  2 D, let P denote the prime of R corresponding to , and let R denote the completion of R with respect to the j j -topology. Put H D R ˝R H . By Proposition 4.4.5, R

R D H  H H e0 D .s/e0 , P where e0 is the idempotent p1n g2G g and s 2 R . Suppose that .r/ 6 P . ThenR rH  C .s/e0  H , and since r is a unit in R , H  rH  C .s/e0 . Thus rH  C H D H .

11.3 The Hopf-Swan Subgroup

247

Now, suppose .r/  P . Note that .s/R 6 P since .r/ and .s/ are relatively prime, and thus s is a unit of R . Thus H D R e0 . It follows that H is the maximal integral order in KP G. Thus H D R e0 ˚ R e1 ˚    ˚ R epn 1 ; where the ei are the minimal idempotents. Consequently, rH  C

R H

D rH  C R e0 Š R e0 ˚ r.R e1 ˚    ˚ R epn 1 / Š H .e0 C r.1  e0 //:

Let ˛ D fa g be the idele in JKG defined as a D 1 for  2 D with .r/ 6 P , a D e0 C r.1  e0 /, for  2 D with .r/  P , and a D 1 for  2 S nD. Then ! \ ˝ R ˛ H a ; r; H D H˛ D KG \ 2S

t u ˝ R ˛ By Proposition 11.3.1, ˝ Rthe˛ Hopf-Swan module r; H gives rise to a class in C.H / that we denote as r; H . The collection of these classes is a subgroup of C.H / called the Hopf-Swan subgroup, which we denote by T .H /. In the case where H D RG, T .H / is the Swan subgroup. We know that R.H /  C.H / and T .H /  C.H /. In fact, for the case H  KCp , D. Replogle and R. Underwood have shown that as required.

T .H /.p1/=2  R.H / (see [RU02, Lemma 2.11]). Moreover, if T .H / is a group of exponent p, then T .H /  R.H /;

(11.4)

and so, if (11.4) holds, then every element of T .H / is a realizable class in C.H /. For this reason, it is important to compute T .H /. The structure of T .H / is the focus of the paper [Un08a], and we review the main results here (largely without proof). For the remainder of this section, we assume that H is an R-Hopf order in KG, where G D Cp . For aRring Y , we let Y  denote the multiplicative group of units of Y . Let q W R ! R= . H / be the canonical surjection, and put  R  W .RH / D R= H =q.R /. R R  R  R  Let W H= H ! R= . H / be the map defined as h C H D .h/ C H .

248

11 The Class Group of a Hopf Order

Lemma 11.3.1. With the notation above, 

R 

T .H / Š W .RH / = H= H =q.R/ . Proof. One has  R   R 

see [RU02, (2.8)] = H= H T .H / Š R= H  R 

 R   Š R= H =q.R/= H= H =q.R /  R 

 =q.R / : Š W .RH / = H= H

t u

By Lemma 11.3.1, there exists a surjection of groups W .R / ! T .H /. H p1 R

Lemma 11.3.2. There is a surjective group homomorphism T .H / ! W

. H/

.

Proof. By Lemma 11.3.1,



R 

T .H / Š W .RH / = H= H =q.R/ .

Let  W W .RH / ! W .RH / denote the homomorphism given as w 7! wp1 . Since  R 

 H= H  q.R /, we have 

R 

p1 p1 T .H /p1 Š W .R / = H= H =q.R / Š W .R / , H

H

t u

which proves the lemma. So, by Lemma 11.3.2, there are surjections q1 ; q2 for which q1

q2

p1

W .R / ! T .H / ! W R : H . H/

(11.5)

We now specialize to the field K D Q.pn /, where n is a fixed integer n  1 and p satisfies Vandiver’s Conjecture. For each integer i , 0  i  p n1 , there exists a Larson order H.i / in KCp (see 5.3). If i D 0, then H.i / D H.0/ D RCp . Our goal is to compute some Hopf-Swan subgroups T .H.i //, including the Swan subgroup T .RC p /. To do this, we need to introduce certain units of R. Definition 11.3.2. Let K C denote the maximal real subfield of K. The cyclotomic units E C of K C are the elements of R generated by 1 and quantities of the form .1a/=2 1

ca D pn

 pan

1  p n

11.3 The Hopf-Swan Subgroup

249

for 2  a  .p n  1/=2, .a; p/ D 1. The cyclotomic units E of K are the elements of R generated by pn and E C . An initial result is the following. Lemma 11.3.3. For each i , 0  i  p n1 , T .H.i // Š W

R

H.i /

.

H.i /

Proof. In view of the sequence (11.5), it suffices to show that the exponent of R

 W R D R= H.i / H.i / =q.R/ H.i /

H.i /

is relatively prime to p  1. Let  D pn  1. By Proposition 5.3.3, H.i / ./

.pn1 i /.p1/

R

H.i /

D

. Since R D Z ˚ Z ˚ 2 Z ˚    ˚ p

.R=./.p i /.p1/ / has order .p  1/p .p n1 Thus .R=./.p i /.p1/ / has the form n1

q

q

n1 .p1/1

n1 i /.p1/1

Z;

as a multiplicative group.

q

1 2  Cpn2      Cpqn1  Cp1 Cpn0  Cpn1

for some integers qi  0, i D 0; : : : ; n  1. The factor Cp1 is identified with the group of units F p. Let hai D F p , and let ca be the corresponding cyclotomic generator. Since pn

ca a mod p, q.R / contains the factor Cp1 , and consequently the exponent n1 of .R=./.p i /.p1/ / =q.R / is relatively prime to p  1. t u Proposition 11.3.2. For each i , 0  i  p n1 , T .H.i // Š .R=./.p

n1 i /.p1/

/ =q.E/:

Proof. In view of Lemma 11.3.3, it suffices to show that q.R / D q.E/. First note that the quotient R =E has order hC .R/ by [Wa97, Theorem 8.2]. Moreover, there exists a surjection R =E ! Q with Q D q.R /=q.E/. Note that both q.E/ and q.R / contain the factor Cp1 in their cyclic decompositions. Consequently, if q.E/ is a proper subgroup of q.R /, then p divides the order of the quotient Q, and hence p divides ŒR W E D hC .K/. By [Wa97, Corollary 10.6], p then divides hC .Q.p //, which is a contradiction since p satisfies Vandiver’s Conjecture. It follows that q.R  / D q.E/. t u Proposition 11.3.2 is a useful mechanism for computation. It says that T .H.i // can be calculated by finding the cyclic decompositions of .R=./.p and q.E/ and then taking the quotient.

n1 i /.p1/

/

250

11 The Class Group of a Hopf Order

To illustrate this, R we set n D 1, i D 0, and consider the R-Hopf order H.0/ D RCp . Then RCp RCp D .p/. It is not hard to show that .R=.p// Š Cpp2  Cp1 : (Prove this as an exercise.) Moreover, the structure of q.E/ has been determined by D. Replogle [Re01], who has proved the following (though not stated exactly in this manner). First, we define the index of irregularity of the prime p is defined as the number s of numerators in the set B0 ; B2 ; B4 ; : : : ; Bp3 that are divisible by p. Proposition 11.3.3. (D. Replogle) Suppose the prime l  3 satisfies Vandiver’s .p1/=2s  Cp1 , where s is the index of irregularConjecture. Then q.E/ Š Cp ity of p. t u

Proof. For a proof, see [Re01, Theorem 1]. Consequently, the Swan subgroup is computed as T .RC p / Š .Cpp2  Cp1 /=.Cp.p1/=2Cs  Cp1 / Š Cp.p3/=2Cs :

For example, take p D 5. Then it is known that the index of irregularity is s D 0, and thus T .RC5 / D C5 . We Rnext consider the case n D 2, i D 0, such that H.i / D H.0/ D RCp and RCp RCp D .p/. We seek to compute the Swan subgroup T .RCp /. In this case, T .RCp / Š .R=.p// =q.E/. We first consider the numerator of this quotient. Proposition 11.3.4. .R=.p// Š Cp2

p2

p 2 3pC3

 Cp

 Cp2 .

Proof. With  D p2  1, observe that R D Z ˚ Z ˚ 2 Z ˚    ˚ p.p1/1 Z; p.p1/

as additive groups. and thus R=./p.p1/ D R=.p/ is isomorphic to Cp Consequently, there are .p  1/p p.p1/1 elements in .R=.p// . The elements 1 C ; 1 C 2 ; 1 C 3 ; : : : ; 1 C p2 have order p 2 in .R=.p// . Moreover, 1 C p1 ; 1 C p ; 1 C pC1 ; : : : ; 1 C p.p1/1 have order p in .R=.p// . For r D 1; 2; : : : ; p  2, .1 C r /p 1 C pr mod .p/;

11.3 The Hopf-Swan Subgroup

251

and so .R=.p// is generated by Cp1 together with the elements 1 C r for r D 1; 2; : : : ; p  2 and the elements 1 C s for p  1  s  p.p  1/  1;

with .s; p/ D 1:

Since there are p  2 values of s with .s; p/ 6D 1, .R=.p// contains p.p  1/  .p  1/  .p  2/ D p 2  3p C 3 copies of Cp in its decomposition. It follows that .R=.p// Š Cp1  Cpp

2 3pC3

p2

 Cp 2 :

t u

The computation of q.E/ is much more complicated, as we shall see. We begin with the following lemmas. Let fca g denote the collection of cyclotomic generators of E. Lemma 11.3.4. Modulo p, the expansions of ca in powers of  D p2  1 are as follows: (i) If a D pm C 1, then 1 ca 1 C mp1 C mp C t2.p1/C1 2.p1/C1 C    C tp.p1/1 p.p1/1 : 2 (ii) If a D pm  1, then 1 ca 1 C mp1 C mp C t2.p1/C1 2.p1/C1 C : : : tp.p1/1 p.p1/1 : 2 (iii) If a D pm C r, r 6D ˙1, then ca r C t2 2 C    C tp.p1/1 p.p1/1 with t2 6 0 mod p. Proof. Note that .a1/.p2 1/=2

ca D p2

.a1/.p2 1/=2C1

C p 2

.a1/.p2 1/=2C.a1/

C    C p 2

;

and write the polynomial fa .x/ D x .a1/.p so that fa .p2 / D ca .

2 1/=2

C x .a1/.p

2 1/=2C1

C    C x .a1/.p

2 1/=2C.a1/

;

252

11 The Class Group of a Hopf Order

To establish the lemma, we compute the Taylor series expansion of fa .x/ in the indeterminate x about the point 1 and reduce modulo p. We seek the coefficients tk such that fa .x/ D t0 C t1 .x  1/ C t2 .x  1/2 C : : : : Let a D pm C r, 0  r  p  1, and put h D .a  1/.p 2  1/=2. We calculate 1 kŠ

tk D

D

X

hC.a1/

i.i  1/.i  2/ : : : .i  .k  1//

i Dh

! i ; k

hCa1 X i Dh

k  0. Now, using the identity n X i k

!

i Dk

we obtain

! nC1 ; D kC1

! ! h hCa ;  tk D kC1 kC1

which yields modulo p 8
if k D 0 if k D 1 .a.a  1/.a C 1// if k D 2. 24

Now, if a 6 ˙1 mod p, then p > 3 and t2 6 0 mod p. Thus the cyclotomic generators ca with a 6 ˙1 have the expansions claimed. We next consider the generators ca , a X and Y be integers, P ˙1 i mod p. LetP X  Y , whose p-adic expansions X D xi p and Y D yi p i , 0  xi ; yi  p  1, satisfy xi  yi for all i . Then a classical result of E. Lucas states that X Y

!

Y xi yi x ;y i

! mod p:

i

An induction argument using Lucas’s formula yields px C x0 py C y0

!

x y

!

x0 y0

! mod p;

(11.6)

11.3 The Hopf-Swan Subgroup

253

which is valid for 0  y0  x0  p  1 and x  y  1. Moreover, if x  y  1 and 0  x0 < y0  p  1, then px C x0 py C y0

! 0 mod p:

(11.7)

Now, with a D pm C 1, ! ! ! ! p.m.p 2  1/=2/ hCa h p.m.p 2  1/=2 C m/ C 1  ;  D kC1 kC1 kC1 kC1 and, for a D pm  1, ! ! ! ! hCa h p.m.p 2  1/=2  p C m/ p.m.p 2  1/=2p/C1  D  : kC1 kC1 kC1 kC1 Thus, by (11.6) and (11.7), 8 0 if 2  k  p  2 ˆ ˆ < m if k D p  1 tk 1 ˆ m if kDp ˆ2 : 0 if p C 1  k  2.p  1/. Thus, the cyclotomic generators ca have the expansions claimed.

t u

Using Lemma 11.3.4, we can compute the orders of the generators of q.E/. Lemma 11.3.5. (i) q.p2 / has order p 2 . (ii) If a D pm C 1, then q.ca / has order p. (iii) If a D pm  1, then q.ca / has order 2p. (iv) Suppose a 6 ˙1, and let the order of a in Cp1 be w. Then w > 2 and q.ca / has order p 2 w. Proof. For (i), observe that p2 D 1 C . The statements (ii), (iii), and (iv) follow from the expansions of Lemma 11.3.4. t u We now proceed with the calculation of q.E/. The group of cyclotomic units E is generated by the set f1; p2 g [ A [ B, where A D fca W 2  a  .p 2  1/=2; .a; p/ D 1; a 6 ˙1g and B D fca W 2  a  .p 2  1/=2; .a; p/ D 1; a ˙1g:

254

11 The Class Group of a Hopf Order

Let h1; p2 ; Ai denote the subgroup of E generated by f1; p2 g [ A, and let hBi denote the subgroup of E generated by B. .p1/=2s

 Cpb  Cp1 , where s is the index Lemma 11.3.6. q.h1; p2 ; Ai/ Š Cp2 of irregularity of the prime p  3 and b is an integer with 0  b  .p 2  4p C 3/=2 C s: Proof. We have q.E/p Š q.E/, Q where qQ W R ! R=./p1 is the canonical surjection. Moreover, q.E/p D q.h1; p2 ; Ai/p since all elements of q.B/ have order either p or 2p by Lemma 11.3.5 (ii), (iii). Hence q.h1; p2 ; Ai/p Š q.E/: Q Now q.E/ Q Š q 0 .E 0 /, where E 0 is the group of cyclotomic units in ZŒp  and q 0 W ZŒp  ! ZŒp =.p/ is the canonical surjection. Thus, by Proposition 11.3.3, q.E/ Q Š Cp.p1/=2s  Cp1 ; and so q.h1; p2 ; Ai/p Š Cp.p1/=2s  Cp1 : .p1/=2s

 Cpb  Cp1 for some integer b. But since Now, q.h1; p2 ; Ai/ Š Cp2 jAj D p.p  3/=2 and q.p2 / has order p 2 , bC

p1 p.p  3/ s  C 1; 2 2

and hence 0b

p 2  4p C 3 C s: 2

t u

Lemma 11.3.7. q.hBi/ Š Cpd  C2 , where d is an integer with 1  d  p  1. Proof. We have jBj D p  1. By Lemma 11.3.5 (ii), (iii), each element in q.B/ has order either p or 2p, and thus q.hBi/ Š Cpd  C2 with 1  d  p  1.

t u

We now have the computation of q.E/. .p1/=2s

Proposition 11.3.5. q.E/ Š Cp2  Cpr  Cp1 , where s is the index of irregularity of the prime p and where r is an integer 1  r  .p  1/2 =2 C s.

11.3 The Hopf-Swan Subgroup

255 .p1/=2s

Proof. By Lemma 11.3.6, q.h1; p2 ; Ai/ Š Cp2 Lemma 11.3.7, q.hBi/ Š

Cpd

 Cpb  Cp1 , and, by

 C2 . Thus .p1/=2s

q.E/ Š Cp2

 Cpr  Cp1 ;

where r is an integer with 1r 

p 2  4p C 3 C s C p  1 D .p  1/2 =2 C s: 2

t u

Finally, we compute T .RCp /. Proposition 11.3.6. Let p  3 be a prime that satisfies Vandiver’s Conjecture, and let K D Q.p2 /. Then .p3/=2Cst

T .RCp / Š Cp2

 Cpp

2 3pC3rC2t

;

where s is the index of irregularity of p and where t satisfies 0  t  minfr; p  2g and is determined by .R=.p// \ q.E/ Š Cp.p1/=2sCt  Cp1 : p

Proof. By Proposition 11.3.2, T .RCp / Š .R=.p// =q.E/. By Proposition 11.3.4, .R=.p// Š Cp2

p2

.p1/=2s Cp 2



Cpr

p2 3pC3

 Cp

 Cp1 , and, by Proposition 11.3.5, q.E/ Š

 Cp1 . Hence p2

T .RCp / Š .Cp2

 Cpp

2 3pC3

p2.p1/=2Cst

Š Cp 2

.p3/=2Cst

Š Cp 2 where t is as claimed.



.p1/=2s  Cp1 /= Cp2  Cpr  Cp1

 Cpp

 Cpp

2 3pC3.rt /Ct

2 3pC3rC2t

; t u

Observe that T .RCp / is a group of exponent p, and hence (11.4) holds; that is, T .RCp / is a subgroup of the realizable classes R.RCp /. Example 11.3.1. One can use a computer algebra system (such as GAP) to compute T .RCp / for certain primes. For p D 3 and p D 5, several computations using GAP yield Table 11.1 below (note that the index of irregularity for these primes is s D 0). 2 Observe that, for p D 5, q.E/ D C25  C57  C4 , and so r D 7. Consequently, t D 0 since T .RC5 / D C25  C56 .

256

11 The Class Group of a Hopf Order Table 11.1 Computation of the Hopf-Swan subgroup p q .h1; ; Ai/ q.hBi/ q.E/ T .RC p / 3 C9  C2 C32  C2 C9  C32  C2 C3 2 2 5 C25  C54  C4 C54  C2 C25  C57  C4 C25  C56

So far in this section, we have computed some Swan subgroups. We next compute some Hopf-Swan subgroups. Let p  3 be a prime that satisfies Vandiver’s Conjecture, and let K D Q.p2 /. Put  D p2  1, so that .p/ D ./p.p1/ . For each integer h i , 0i  i  p, there exists a Larson order H.i / in KCp , given as . In this section, we compute the Hopf-Swan subgroups T .H.p//, H.i / D R g1 i T .H.p  1//, and T .H.p  2//. We begin R with

some observations. for i D p; p  1; p  2; : : : ; 2; 1; 0, we have H.i / H.i / D ./.pi /.p1/ . Let qi W R ! R=./.pi /.p1/ be the canonical surjections, each playing the role of q in the computation of the Swan subgroup. Using (11.5), it is not difficult to prove the following. Proposition 11.3.7. T .H.p// D 1. t u

Proof. Exercise. We shall compute T .H.p  1// and T .H.p  2// by using the formula T .H.i // Š .R=..pi /.p1/ / =qi .E/

for i D p  1; p  2, which follows from Proposition 11.3.2. We have the following lemma. Lemma 11.3.8. For i D p  1; p  2; p  3; : : : ; 2; 1, pi 1

.R=./.pi /.p1/ / Š Cp2

 Cp.p2/C.pi 1/.p3/  Cp1 :

Proof. We have R D Z ˚ Z ˚ 2 Z ˚    ˚ p.p1/1 Z: For p  2  i  p  1, .R=./.pi /.p1/ / has order .p  1/p .pi /.p1/1 as a multiplicative group. There are p  i  1 copies of Cp2 in the cyclic decomposition of .R=./.pi /.p1/ / . Let N be the number of copies of Cp that occur in the cyclic decomposition of .R=./.pi /.p1/ / . Then N C 2.p  i  1/ D .p  i /.p  1/  1, so that N D .p  i /.p  1/  1  2.p  i  1/ D .p  2/ C .p  i  1/.p  3/:

11.3 The Hopf-Swan Subgroup

257 pi 1

Thus .R=./.pi /.p1/ / Š Cp2

.p2/C.pi 1/.p3/

 Cp

 Cp1 .

t u

Proposition 11.3.8. Suppose p  3 satisfies Vandiver’s Conjecture. .p3/=2Cs T .H.p1// Š Cp , where s is the index of irregularity of p.

Then t u

Proof. Exercise.

Finally, we compute T .H.p  2//. We already know the numerator of T .H.p  2// by Lemma 11.3.8, so it is a matter of computing qp2 .E/. Again we use the fact that E is generated by the set f1; p2 g [ A [ B. 

Lemma 11.3.9. qp2 .h1; p2 ; Ai/ Š Cp2  Cp  Cp1 , where   .p  3/=2  s. Proof. Since p2 D 1C, qp2 .p2 / has order p 2 . By Lemma 11.3.4 (iii), qp2 .ca /, a 6 ˙1, has order pw with 2p < pw  p.p  1/. Thus qp2 .h1; p2 ; Ai/ Š Cp2  Cp  Cp1 for some   0. Observe that qp1 .h1; p2 ; Ai/ D qp1 .E/ since cpm˙1 ˙1 mod ./p1 by Lemma 11.3.4 (i), (ii). Moreover, by Proposition 11.3.8, qp1 .E/ Š Cp.p1/=2s  Cp1 ; and hence qp1 .h1; p2 ; Ai/ Š Cp.p1/=2s  Cp1 : But this says that .p  1/=2  s   C 1, and thus   .p  3/=2  s.

t u

Lemma 11.3.10. qp2 .hBi/ Š Cp  C2 . Proof. We observe that, for integers m; n, 1  m; n  .p  1/=2, cpmC1 cpnC1  cp.mCn/C1 0

pmC1

1  p 2 pm=2

D @p 2

1  p 2

10 A @

p.mCn/=2

.1  p2 /2

p.mCn/=2

.1  p2 /2

D p 2

D p 2 



pnC1

1  p 2

p.mCn/=2

D p 2

pnC1

1  p 2

p2

1

pmCpnC1

A   p.mCn/=2 2 p

pmCpnC2

C p2

pmCpnC1

 1 C p2

.1  p2 /2 p2 .1  p2 /.1  p2 /

0 mod ./2.p1/ :

1  p 2

1  p 2



pnC1 pmC1 pmCpnC2 1  p 2   p 2 C  p 2

pmCpnC1 .1  p2 / 1  p2

pmC1

 p 2

1  p 2 pn=2

mp

np

pmCpnC2

C  p 2  p 2



258

11 The Class Group of a Hopf Order

Thus i cpC1 cpi C1 mod ./2.p1/

for i D 1; : : : ; p  1. Moreover, cpi C1 cp.pi /1 mod ./2.p1/ ; and thus, for i D .p  1/=2 C 1; : : : ; p  1, i cp.pi /1 mod ./2.p1/ : cpC1

It follows that qp2 .hBi/ is generated by the two elements qp2 .cpC1 / and qp2 .1/. Since these elements have order p and 2, respectively (use Lemma 11.3.4), we conclude that qp2 .hBi/ Š Cp  C2 :

t u



Lemma 11.3.11. qp2 .E/ Š Cp2  Cp  Cp1 , where   .p  3/=2  s. 

Proof. By Lemma 11.3.9, qp2 .h1; p2 ; Ai/ Š Cp2  Cp  Cp1 , with   .p  3/=2  s, and, by Lemma 11.3.10, qp2 .hBi/ Š Cp  C2 . Thus qp2 .E/ Š Cp2   Cp  Cp1 , where   .p  3/=2  s. t u Proposition 11.3.9. Suppose p  3 satisfies Vandiver’s Conjecture. Then T .H.p  2// Š Cp ; with .p  3/=2 C s    .3p  7/=2 C s, where s is the index of irregularity of p. Proof. We have T .H.p2// Š .R=./2.p1/ / =qp2 .E/. Now .R=./2.p1/ / Š  Cl 2  Cl2l5  Cl1 by Lemma 11.3.8, and qp2 .E/ Š Cp2  Cp  Cp1 by Lemma 11.3.11. Hence T .H.p  2// Š .Cp2  Cp2p5  Cp1 /=.Cp2  Cp  Cp1 / Š Cp2p5 Š Cp ; with 0    .3p  7/=2 C s. Also, in view of the surjection T .H.p  2// ! T .H.p  1//,  is bounded below by .p  3/=2 C s. t u For p  5, j D 1; 2, T .H.p  j // is a non-trivial p-group. Thus (11.4) holds and there is a non-trivial class .M / in R.H.p  j //. By Proposition 11.2.4, .M / is the class of a ring of integers S in a field extension L=K with group Cp . Moreover, since .S / is non-trivial, S is a semilocal Galois H.p  j /-extension that is not free over H.p  j /.

11.4 Chapter Exercises

259

11.4 Chapter Exercises Exercises for 11.1 1. Show that there exists an adele fa g in VQ for which C.fa g/ > 1. 2. Let fa g be an idele in VQ for which C.fa g/ > 1. Prove that ja j D 1 for almost all . 3. Prove Proposition 11.1.2, parts (ii) and (iii). 4. Prove Proposition 11.1.4. 5. Let R be the ring of integers in an algebraic number field. Prove that R is a PID if and only if C.R/ D 1. p p 6. Find anp example of a locally free ZŒ 5-submodule of Q. 5/ that is not free over ZŒ 5. Exercises for 11.2 7. Let n  1 be an integer, and let H be an R-Hopf order in KCpn , pn 2 K. (a) Let M be a locally free H -module of rank k. Prove that M is a free H -module of rank k if and only if the class of h.M / is trivial in C.H /. (b) Let M and N be locally free H -modules that satisfy M ˚ Hm Š N ˚ Hn for some m; n. Show that .h.M // D .h.N // in C.H /.

Chapter 12

Open Questions and Research Problems

In the chapters of this book, we have developed some of the central themes in the study of Hopf algebras and Hopf orders. As one might expect, there are many outstanding problems in this field that have not been solved. We now revisit some of the topics in this book and give an account of some open questions and research problems.

12.1 The Spectrum of a Ring Problem 1. In the manner illustrated in Figures 1.1, 1.2, and 1.3, find the structure of the associated map Spec ZV ! Spec Z induced from the ring inclusion Z ! ZV , where V denotes the Klein 4-group. Problem 2. Find the structure of the associated map Spec ZCpn ! Spec Z induced from the ring inclusion Z ! ZCpn .

12.2 Hopf Algebras Problem 3. A major open problem is the following. Let p be a prime, and let n  1 be an integer. Let K be a finite extension of Qp , and let Cpn denote the cyclic group of order p n . Give a classification of all of the Hopf orders in KG (More on this is presented below). Problem 4. Let K be a finite extension of Qp . It has been conjectured that every Hopf order in KCpn can be determined by n valuation parameters and n.n  1/=2 unit parameters. Prove or disprove this conjecture. (Certainly, the conjecture is true for the cases n D 1; 2. For n D 3, no Hopf order in KCp3 has yet been constructed that requires more than six parameters.)

R.G. Underwood, An Introduction to Hopf Algebras, DOI 10.1007/978-0-387-72766-0 12, © Springer Science+Business Media, LLC 2011

261

262

12 Open Questions and Research Problems

12.3 Valuations and Larson Orders Problem 5. There are various problems regarding the counting of order bounded and p-adic group valuations on finite groups (see for instance the paper of A. Koch and A. Malagon [KM07]). Let K D Q.pn /. Compute the number of p-adic obgvs on Cpn and Cpn . More generally, compute the number of p-adic obgvs on a p-group of order p n . Problem 6. R. Larson [Lar76] has given the following generalization of group valuations. Let W Z>0 ! Z be a function defined as  .n/ D

p 1

if n D p a , for some a otherwise.

Let G be a finite Abelian group, let K be a finite extension of Q, and let IK denote the group of fractional ideals in K. A global group valuation is a function  W G ! IK [ f0g that satisfies (i) .g/  R, .g/ D f0g, if and only if g D 1; (ii) .gh/  .g/ C .h/. A global group valuation  is order bounded if .jgj/R  .g/.jgj/ : An order-bounded global group valuation (obggv)  is p-adic if .g

.jgj/

/  .g/

.jgj/

:

Let H be an R-Hopf order in KCp . Let P be a prime ideal of R. By Proposition 7.1.2, the R -Hopf order HR in KP Cp is a Larson order and is given by a p-adic obgv on Cp . Is H determined by a p-adic obggv on Cp ?

12.4 Hopf Orders in KCp2 Problem 7. Let p be prime and let K be a finite extension of Qp . In 8.4, we showed that every Hopf order in KCp2 for p D 2; 3 can be written in the form   p g  1 gu  1 ; H DR i j for some u D u0 e0 C u1 e1 C    C up1 ep1 , um 2 K, where em are the idempotents in Khg p i. It would be of interest to extend this result to primes p > 3 for then one could give an alternate (perhaps shorter) proof of the valuation condition for n D 2.

12.5 Hopf Orders in KCp3

263

12.5 Hopf Orders in KCp3 Problem 8. As we indicated above, a major problem is to complete the classification of R-Hopf orders in KCp3 . Can every Hopf order in KCp3 be written as a circulant matrix Hopf order or the linear dual of such a Hopf order? Problem 9. Does every circulant matrix Hopf order satisfy the valuation condition for n D 3? Does an arbitrary Hopf order in KCp3 satisfy the valuation condition for n D 3? These two questions are important for the following reason. If the Hopf order H in KCp3 , hgi D Cp3 , satisfied the valuation condition for n D 3, and induced the short exact sequence E W R ! A.i; j; u/ ! H ! H.k/ ! R; with pk  „.H /.g p /, then there would be a short exact sequence of Hopf orders E0 W R ! A.i; j; u/ ! H0 ! H.k/ ! R; where H0 is an R-Hopf order in K.Cp2  Cp / of the form " 2 # gp  1 gp au  1 g  1 ; ; : H0 D R i j k The Baer product E  E01 D E 0 would then yield an extension E 0 W R ! A.i; j; u/ ! H 0 ! H.k/ ! R; which over K would appear as K ! KCp2 ! KCp3 ! KCp ! K: Thus the classification of Hopf orders in KCp3 would reduce to the case K.Cp2  Cp /, presumably a simpler problem. Of course, all of this is predicated on giving a precise diagramwise definition of the Baer product of two extensions of A.i; j; u/ by H.k/ similar to the way we defined the Baer product in 8.3. Problem 10. The construction of formal group Hopf orders in KCpn (Chapter 6) when applied to the case n D 3 yields a collection of R-Hopf orders in KCp3 . What is the structure of the linear duals of these Hopf orders? Also, what is the precise relationship between the formal group Hopf orders in KCp3 , the duality Hopf orders of 9.1, and the circulant matrix Hopf orders of 9.2? Also, it would be of interest to compute the Larson order A.„.H //, where H is one of these Hopf orders in KCp3 .

264

12 Open Questions and Research Problems

Problem 11. Another approach to the classification of Hopf orders in KCp3 would be to extend the method of 8.4 to show that every Hopf order in KCp3 could be written in the form # " 2 gp  1 gp av  1 gu  1 R ; ; i j k for some u 2 KCp2 .

12.6 Hopf Orders and Galois Module Theory Problem 12. Let p  5 be prime, and let K .p/ be the splitting field of the polynomial hp .x/ D fp .x/=.x C 1/ defined in 10.1. By Proposition 10.1.10, the Galois group of K .p/ is Cp1 . What are the equivalence classes of extensions of j j1 to K .p/ ? For q prime, what are the equivalence classes of extensions of j jq to K .p/ ? Problem 13. Compute the ring of integers R .p/ of K .p/ for primes p  5. Compute disc.R.p/ =Z/. Which primes q 2 Z ramify in K .p/ ? Problem 14. Let R.p/ be as in Problem 14. Let Z ! R.p/ be the ring inclusion. Illustrate the spectral diagram for the associated map W Spec R.p/ ! Spec Z. Problem 15. Let K be a Galois extension of Q with Abelian Galois group G and ring of integers R. Since R is Dedekind,there exists a unique factorization disc.R=Z/ D .p1 /e1 p2e2    .pk /ek for primes pi 2 Z. By Proposition 10.2.14, these are precisely the primes that ramify in R. Let q be a prime of Z that is unramified in R, let Q be a prime of R lying above q, and let Q be the Frobenius element at Q. For a prime Q 0 lying above q, one has Q0 D Q . Thus we may define a function FK=Z W Spec Znfp1 ; p2 ; : : : ; pk ; !g ! G by the rule FK=Z .q/ D Q , where Q is a prime lying above q. Proposition 12.6.1. Let K be a Galois extension of Q with Abelian Galois group G and ring of integers R. For each element  2 G, the primes q 2 Z with FK=Z .q/ D  1 have density jGj . Proposition A is known as the Tchebotarev Density Theorem for Abelian Extensions (TDT). The TDT implies that the density of primes of Z that completely split in ZŒi is 1=2. Use a computer algebra/graphics program to extend the spectral diagram in Figure 1.1 to the first 1000 primes of Z. Show that this empirical data is consistent with the Tchebotarev density of 1=2.

12.7 The Class Group of a Hopf Order

265

Let K .p/ be the number field as in Problem 13, and let Z ! R.p/ be the ring inclusion. The TDT implies that the density of primes of Z that completely 1 split in R.p/ is p1 . Construct the spectral diagram for the associated map W .p/ Spec R ! Spec Z. Include at least 1000 primes of Z. Show that this empirical 1 data is consistent with the Tchebotarev density of p1 . Problem 16. Let Spec ZC3 ! Spec Z be the map illustrated in 1.3. Use a computer algebra/graphics program to extend the spectral diagram in Figure 1.3 to the first 1000 primes of Z. What does this suggest about the ratio of primes in Z that factor as P1 P2 (as does 11) to primes in Z that factor as P1 P2 P3 (like 13)? Is there an analog of the TDT for rings that are not rings of integers in number fields?

12.7 The Class Group of a Hopf Order p Problem 17. Let K D Q. d / be a real quadratic extension with d > 0 and squarefree, let S D fg be the set of normalized absolute values on K, and let JK denote the group of ideles over K. Find a constant such that for an adele fa g 2 JK there exists an element b 2 K C , b 6D 0, for which .b/  .a /; 8 2 S . Problem 18. The group of units in an algebraic number field can be determined using the following result of Dirichlet. Dirichlet’s Unit Theorem. Let K be a finite extension of Q of degree n. Then U.R/ Š ZrCs1  W; where r is the number of embeddings of K into R, s is the number of pairs of conjugate embeddings of K into C, and W is the subgroup of R generated by the roots of unity in R. Moreover, n D r C 2s. Let p  5 be prime, and let K .p/ be the splitting field of the polynomial hp .x/ D fp .x/=.x C 1/. Since every root of hp .x/ is real, one has  Z U.R.p/ / Š h1i  Z …: „  Zƒ‚ p2

Find a system of fundamental units for R.p/ . (We were very close to giving a proof of Dirichlet’s Unit Theorem using the theory of adeles developed in Chapter 11. See [CF67, Chapter II, 18].) Problem 19. For the prime p D 7, use a computer algebra system (like GAP) to compute the Hopf-Swan subgroup for each Larson order H.i / in K D Q.49 /. Problem 20. Let n  1 be an integer and let K D Q.pn /. Compute the structure of T .H / for a Larson order H in KCp . Let K D Q.p2 / and let H be an R-Hopf order in KCp2 . Compute T .H /.

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Index

A absolute value, 95 archimedean, 100 normalized, 234 discrete, 98 normalized, 234 equivalence, 95 extension, 95 p-adic, 14 adele ring, 236 affine group scheme, see R-group scheme associated order, 229 augmentation ideal, 44

B Baer product, 168 Byott, N., viii, 142, 223, 224, 245

C Cassels, J. W. S., viii category, 23 axioms for a, 23 coproduct, 30 Cauchy sequence, 96 Childs, L., viii, 119, 124, 129, 181 class group, 234 of a Hopf order, 242 class invariant map, 245 class number, 239 closure, 21 coassociative property, 55 coboundary, 149 cocycle, 148 coinverse map, 39 coinverse property, 55

complete tensor product, 122 completion, 96 comultiplcation map, 39 content map, 237 kernel of, 237 counit map, 39 counit property, 55 crossed product algebra, 213 cyclotomic units, 248, 249

D decomposition group, 205 Dedekind domain, 66 different, 207, 209 Dirichlet’s Unit Theorem, 265 discrete valuation, 98 discriminant, 88, 207, 209 relative to a Hopf order, 219 dual basis, 61 dual module, 86

E epimorphism, 50 exact sequence, 47 extension, 148 Galois, 197 of absolute values, 95 of group schemes, 156 equivalent extensions, 156 generically trivial, 164 of groups, 148 equivalent, 148 tame, 226 wild, 226

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272 F faithfully flat ring homomorphism, 48 Fermat’s Two Squares Theorem, 4 flat ring homorphism, 48 formal group, 115 additive, 116 commutative, 115 endomorphism, 120 homomorphism of, 119 isogeny, 120 linear isomorphism of, 115 multiplicative, 116 n-dimensional, 115 generically split, 118 polynomial, 115 Fr¨ohlich, A., viii, 240 fractional ideal, 67 invertible, 233 functor, 27 covariant, 27 natural transformation, 27 representable, 29 G Galois H -extension, 215 semilocal, 244 Galois extension, 197 of rings, 214 Galois group, 197 Gauss sum, 183 generating integral, 80 Greither, C., 137, 138, 155, 165 group of automorphisms, 197 group valuation, 100 global, 262 order bounded, 103 order-bounded p-adic, 103 H H -extension, 215 Galois, 215 H -module locally free rank one, 241 Hopf algebra, 55 antihomomorphism, 60 cocommutative, 57 coinverse map, 55 comultiplication map, 55 counit map, 55 homomorphism of, 58 Hopf comodule, 70 Hopf module, 71

Index Hopf order, 82 circulant matrix, 194 duality, 184 formal group, 125 Greither order, 168 largest Larson order, 111 Larson order, 105 realizable, 223 trivial, 82 Hopf, H., vii Hopf-Swan module, 246 Hopf-Swan subgroup, 247 I ideal group, 235 idele group, 236 index of irregularity, 250 inertia group, 206, 208 integral, 57 integrally closed domain, 66 K kernel, 47 Koch, A., 104, 262 L Larson, R., viii, 70, 73, 77, 88, 95, 104, 142, 262 linear dual, 61 localization, 2 M Malagon, A., 104, 262 McCulloh, L., 244, 245 module algebra, 215 fixed ring of a, 215 N natural transformation, 27 coboundary, 160 cocycle, 156 Nichols, W., viii nilpotent, 2 nilradical, 9 Noether’s Theorem, 228 Noether, E., 228 Noetherian ring, 65 norm map, 198 of an extension, 237 relative, 208 Normal Basis Theorem, 227

Index O Oort, F., viii order with respect to a prime, 98

P p-adic integers, 112 partially ordered set, 1 chain in a, 1 presheaf, 23 of ordinary functions, 23 pth power map, 52

Q quotient sheaf, 50

R R-group scheme, 39 additive, 40 homomorphism of, 41, 43 multiplicative, 40 group of the nth roots of unity, 41 special linear, 41 trivial, 39 R-order, 82 ramification index, 203 relative, 207 ramified prime, 207 rational function, 11 realizable classes, 244 in C .H /, 245 Replogle, D., 247, 250 representable group functor, see R-group scheme

S semilocal ring, 244 sheaf, 25 short exact sequence, 50 of Hopf algebras, 61 of Hopf orders, 85 smash product, 215 spectrum, 2 associated map, 3 generic point, 22 points of, 2 splitting field, 196 structure, 25 Swan subgroup, 247 Sweedler notation, 35 Sweedler, M., viii, 70, 73, 77

273 T tame, 226 tamely ramified, 207 Tate, J., viii Tchebotarev Density Theorem, 264 Teichm¨uller character, 130 topological space, 13 basis for a, 17 compact, 20 compact subset, 20 continuous map, 16 Hausdorff, 19 homeomorphic spaces, 17 homeomorphism, 16 irreducible, 22 open covering of, 20 reducible, 22 topology, 13 closed set in a, 13 discrete, 16 I -adic, 122 open set in a, 13 p-adic, 14 product, 15 restricted, 16 quotient, 15 standard, 13 subspace, 15 Zariski, 15 trace map, 198 relative, 208 with respect to decomposition group, 206 twist map, 36

U Underwood, R., 119, 124, 142, 181, 185, 247 uniformizing parameter, 67 unramified prime, 207

V valuation condition, 142 for n D 3, 185 Vandiver’s Conjecture, 239 virtual characters, 240

W wild extension, 226

Z Zorn’s Lemma, 1