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Non-Additive Ring and Module Theory IV the Brauer Group of a Symmetric

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Non-Additive Ring and Module Theory IV the Brauer Group of a Symmetric Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 549 Brauer Groups Proceedings of the Conference Held at Evanston, October 11-15,1975 * 416 109 546 200 16 Edited by D. Zelinsky Springer-Verlag Berlin · Heidelberg · NewYork1976 Editor Daniel Zelinsky Northwestern University Department of Mathematics Evanston, II. 60091/USA Library of Congress Cataloging in Publication Data Main entry under title: Brauer groups. (Lecture notes in mathematics ; 5^9) "Sponsored by Northwestern University." Bi bli ography: ρ. Includes index. 1. Brauer group--Congresses. 2. Separable algebras—Congresses. I. Zelinsky, Daniel. II. Northwestern University, Evanston, 111. III. Series: Lecture notes in mathematics (Berlin) ; 5^9. QA3.L28 no. 5^9 [QA251.3] 510'.8s [512».2*0 76-kekie AMS Subject Classifications (1970): 13A20, 16A16, 18H20, 14C20, 14H99, 14L15, 18D10 ISBN 3-540-07989-0 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07989-0 Springer-Verlag New York · Heidelberg · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re­ printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. CONTENTS Lindsay N, Childs On Brauer groups of some normal local rings..... 1 P. R. DeMeyer The Brauer group of affine curves 16 M.-A. Knus, M. Ojanguren, D. J. Saltman On Brauer groups in characteristic ρ 25 Gerald S. Garfinkel A module approach to the Chase-Rosenberg-Zelinsky sequences 50 Daniel Zelinsky Long exact sequences and the Brauer group 6? Andy R. Magid The Picard sequence of a fibration 71 George Szeto The Pierce representation of Azumaya algebras 86 William Gustafson A remark on class groups 92 Philip LaFollette Splitting of Azumaya algebras over number rings 100 H. F. Kreimer, Jr. Abelian p-extensions and cohomology 104 Bodo Pareigis Non-additive ring and module theory IV. The Brauer group 112 of a symmetric monoidal category Morris Orzech Brauer groups of graded algebras 134 Stephen U. Chase On a variant of the Witt and Brauer group l48 IV The Conference on Brauer Groups was originally titled Conference on Brauer and Picard Groups. The present title is more nearly repre• sentative of the contents of the conference and these proceedings. The conference was sponsored by Northwestern University and was held there (Evanston, Illinois) from October 11 to 15, 1975. The list of participants which follows gives the university of each participants at the time of the conference (Department of Mathe• matics in each case). Professor Chase had to cancel his attendance but kindly submitted his manuscript for these Proceedings. Besides the papers published here, the following were read: R. T. Hoobler How to construct U, P. D.'s M. Ojanguren Generic splitting rings S. Rosset Some solvable group rings are domains D. Haile Generalization of involution for central simple algebras of order m in the Brauer group S. A. Amitsur Cyclic splitting of generic division algebras G. Szeto Lifting modules and algebras CONFERENCE ON BRAUER GROUPS List of Participants AMITSUR, S. A. LAFOLLETTE, Philip Hebrew University Indiana Univ. Jerusalem, Israel Bloomington, Indiana AUSLANDER, Bernice LEE, Hei-Sook Univ. of Massachusetts Queens University Boston, Massachusetts Kingston, Ontario CHASE, Stephen U. MAGID, Andy R. Cornell University Univ. of Illinois Ithaca, New York Urbana, Illinois CHILDS, Lindsay N. OJANGUREN, Manuel SUNY at Albany Westfälische-Wilhelms Albany, New York ..University Munster, Germany COOK, P. M. Michigan State Univ. ORZECH, Morris East Lansing, Michigan Queens University Kingston, Ontario DEMEYER, Frank R. Colorado State Univ. PAREIGIS, Bodo Ft. Collins, Colorado Ludwig-Maximilians University ELGETHUN, Edward Munchen, Ge rmany Univ. of N. Florida Jacksonville, Florida REINER, Irving Univ. of Illinois GARFINKEL, Gerald S. Urbana, Illinois New Mexico St. Univ. Les Çruces, New Mexico ROSSET, Shmuel Tel Aviv University GUSTAFSON, William Tel Aviv, Israel Indiana University Bloomington, Indiana SALTMAN, David J. Yale University HAILE, Darrell New Haven, Conn. Northwestern Univ. Evanston, Illinois SMALL, Charles Queens University H00BLER, Raymond Kingston, Ontario City College, CUNY New York, New York SZETO, George Bradley University INGRAHAM, Edward Peoria, Illinois Michigan State Univ. East Lansing, Michigan ZELINSKY, Daniel Northwestern Univ. KNUS, Max-Albert Evanston, Illinois ΕΤΗ Zurich, Switzerland KREIMER, Η. F. Florida State Univ. Tallahassee, Florida NON-ADDITIVE RING AND MODULE THEORY IV The Brauer Group of a Symmetric Monoidal Category Bodo Pareigis In [5] >[6] and [7] we introduced general techniques in the theory of a monoidal category, i.e. of a category C with a bifunctor H: C χ C > C , an object I ε C and natural isomorphisms α: Α Η (B a C) > (Α Η Β) Η C 5 λ : I Η A = A and 5 : A E I = A which are coherent in the sense of \j>, VII. 2] . In this paper we want to in - troduce the notion of a Brauer group of C . For this purpose we are going to assume that C is symmetric, i.e. that there is a natural isomorphism γ: Α Β Β = Β s A which is coherent with α, λ and j> [3]. One of the main models for such a category C is, apart from the category of k-modules for a commutative ring k , the dual of the cate - gory of C-comodules for a cocommutative coalgebra C . This category is a symmetric monoidal category, but it is not closed. Another type of monoidal categories, which are not symmetric but which allow the construction of Brauer groups, are for example catego - ries of dimodules over a commutative, cocommutative Hopf algebra [2] . Their general theory will be discussed elsewhere. In many special cases of symmetric monoidal categories the basic ob ject I turns out to be projective, i.e. the functor C(i,-) preserve epimorphisms. In the general situation, however, it turns out that ther may be constructed two Brauer groups B^(C) and B2(C) and a group-ho momorphism B2(C) > B^CC) , which is an isomorphism if I projective We will construct these two Brauer groups and discuss under which co dition for a functor F : C > V we get an induced homomorphism B.(F): B.(C) > B±(V) . 113 Preliminaries In [7] we proved analogues of the Morita Theorems which will be used in this paper. For the convenience of the reader we will collect the most important definitions and facts of [5], [6] and [7]. If Ρ is an object of C we denote by P(X) the set C(X5p) for Χ ε C . Elements in Ρ H Q (X) will often be denoted by ρ Η q . If the functor C(P Η -,Q) is representable then the representing object is [P,Q] , so that C(P Η X,Q) = C(X, [P,Q] ) = [P,Q] (X) . The "evalua­ tion" P(X) χ [P,Q](Y) > Q(X Η Υ) , induced by the composition of morphisms, is denoted by P(X) χ [P,Q](Y)3 (p,f) '—» <p>f ε Q(X κ Y) . Thus the "inner morphism sets" [P>Q] operate on Ρ from the right. In [5, Proposition 3.2] we prove that any natural transformation P(X) > Q(X Η Υ) , natural in X , is induced by a uniquely determined element of [P,Q] (Υ) , if [P,Q] exists. We call an object Ρ ε C finite or finitely generated projective if [P,l] and [P,P] exist and if the morphism [Ρ,ΐ] κ Ρ > [p>p] in - duced by P(X) χ [P,l](Y) χ P(Z) ^ (p,f,p') «—> <P>fp' εΡ(ΧΗΥκΖ) is an isomorphism. For [Ρ,ΐ] κ Ρ > [p>p] to De an isomorphism it is necessary and sufficient that there is a "dual basis" f s Ρ0 ε [P,l] ss P(I) such that <p> fQpQ = ρ for all ρ ε P(X) and all Χ ε C . The difference between finite and finitely generated pro - jective objects, as discussed in [8] , does not appear here. A finite object Ρ is called faithfully projective if the morphism Ρ Η [Ρ,ΐ] £ I y induced by the evaluation, is an isomorphism. This ^ r π is equivalent to the existence of p. H f1 ε Ρ a (^) with I[P,P] 1 [P,P] <P1>f1 = 1 ε 1(1) . If there exists an element ρ Η f1 ε Ρ Ε [Ρ,ΐ] (I) with <P^>fi = 1 λ then Ρ is called a progenerator. Now Ρ Η [Ρ,il * Ρ « [P,l] is an epimorphism; if I is projective, then Ρ is faithfully projective iff Ρ is a progenerator. Let .C denote the category of Α-objects in C with A a monoid. 114 X ε C with P a C 3 1 P Η LEFT B RI HT A Then a functor A X > Α B B A S biobject is a category equivalence iff βΡ is faithfully projective and A = g[P,P] as has been proved in [7]. For this Morita equivalence all the usual conclusions hold, in particular the centers a AND A[ >A]A B |J3,Β]b of A resp. Β are isomorphic monoids if they exist. The Brauer group Β {C) . Let C be a symmetric monoidal category. A monoid A in C is called 1-Azumaya if C s X 1 » A Η Χ ε C is an equivalence of cate - gories.
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