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
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© 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) '—»
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') «—>
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
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]
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. Thus the Morita Theorems, in particular [7, Theorem 5. Ï] can be applied.
Proposition 1 : A monoid A is 1-Azumaya iff Α ε C _is faithfully pro jective and
ψ: A Η A (X)3 a Β b 1 > (A(Y) 3 c 1 > acb ε A(X β Υ) ) ε [Α,Α] (Χ)
is an isomorphism.
Proof: Let A be faithfully projective and ψ be an isomorphism. Let
A°P be the monoid on A with inverse multiplication. Then
ψ: Aop H A » CA>A1 is an isomorphism of monoids. Thus the categories
ACA " ^ [A A] are eQuivalent with the functor Μ 1 ? M . Furthermore
C 3 Χ ι } A H Χ ε C |-A Aj is an equivalence by [7, Theorem 5. *G hence
C 3 Χ ι > A H Χ ε ACA is an equivalence and A is 1-Azumaya.
Conversely if A is 1-Azumaya the morphism ψ which exists for any monoid A induces a commutative diagram C > C 115
Hence Α ε C is faithfully projective in C by [7, Theorem 5-1} and so ψ must induce a category isomorphism and must even be an isomor phism [7, Theorem 5.1 d)} . Q.E.D.
Recall that the center of a monoid A is the object a[A,A}a , if
it exists. a[A3A]a is the representing object of the functor
C ε X » >A CA(A a Χ,Α) ε S . Observe that for this definition we need the symmetry of the monoidal category C .
A A Let A be a monoid in C . Since we have A[ > ]A(X) £ A(X) , the
A A inclusion given by the isomorphism A[ > ] = A using the multiplica - tion with A from the right, it is easy to see that
a ε a[A,A]a(X) iff ab = ba for all b ε A(Y) and all Υ ε C . since ab = ba for all a ε Im(^(X): I(X) >A(X)) and all b ε A(Y) , all
Χ,Υ ε C , we get that ^(X) maps I(X) into a[A,A}A(X) . If this mor - phism is an isomorphism then A is called a central monoid.
Let A be 1-Azumaya. Then [A,A] is Morita equivalent to I hence the center I of I coincides with the center of [A,A] via the mor - phism : I * [A>A] \j> Corollary 6.3] . Thus the morphism
I A A°a A J-> [A,A] is injective. f is defined by f (a) = 1 H a . Hence I(X) is con -
A A tained in the center A[ s }A(X) · Now let a ε A(X) such that ab = ba for all b ε A(Y) , all Υ ε C . Then (1 a a)(b a c) = b a ac = b a ca = op (b a c)(l a a) for all b a c ε A a A(Y) . Thus
Corollary 2 : If A is 1-Azumaya then A is central.
Proposition 3: Let A , Β be 1-Azumaya then A a Β is 1-Azumaya. 116
Proof: Let f Β a resp. g H b be a dual basis for A resp. Β ,\ ο ο °o O ι
Then fQ H go H aQ Η bQ ε [Α Β Β,ΐ] E A Η Bei) is a dual basis for Α κ Β where we identified [A,l] E QB,I] with [A Β B,lJ . Further - more we have QA,A] Β [Β,Β] = [Α Β [Α,Ι] , [Β,Β]] = [Α Ε Β Β [Α,Ι],Β] = [Α Β Β , Α Β Β] since A and Β are finite [8, Theorem 1.2] . Hence [Α Β Β, I] and [Α Β Β , Α Β Β] exist, op Let A : Α Η A and Α* ε C be the dual [A,l] of A . With AE I the analogous notation for Β we get
(Α Β Β) Β (Α1 Ε Β') = (AB A*) a (Β Β Β1) = I AE Β BE AE BE since A , Β are faithfully projective. Hence Α Ε Β is faithfully projective in C .
Finally since [Α,Α] Β [Β,Β] - [Α Β Β , Α Β Β] we get that
ψ: ABBBABB > [Α Β Β , Α Β B] is an isomorphism.
Proposition 4 : Let Ρ be faithfully projective then JjP,is
1-Azumaya.
Proof: We know that [Ρ,ΐ] Ε Ρ - [Ρ,Ρ] as [P5P] - [Ρ,Ρ] " objects. Furthermore C 3 Χ ι > [Ρ,ΐ] Β Χ ε C and C 3 Χ ι—^ Χ Β Ρ ε C are equivalences. Hence
C 3 Χ ι—> [Ρ,Ρ] Β X = [Ρ,ΐ] Β Χ Β Ρ ε C is an equivalence, [Ρ,Ρ] [Ρ,Ρ] since C s> Υ ι—> Y Β Ρ ε C is also an equivalence. [Ρ,Ρ] [Ρ,Ρ] [Ρ,Ρ]
Proposition 5: Let Ρ , Q be faithfully projective, then Ρ Β Q is faithfully projective and [Ρ,Ρ] Β [Q,Q] - [Ρ Β Q , Ρ Β θ] as monoids.
Proof: Since Ρ and Q are finite we get for all Χ, Υ ε C that f: [Ρ,Χ] E [Q,Y] 3 f Ε g ι > (ρ Β q ι ) <ρ> f Β g)c[p Β Q, Χ Β Y] is an isomorphism and the right side exists. In particular 117
[Ρ a Q,l] and [ρ s Q,P a Q] exist. Furthermore we have
f(f a gty(f'a g) =
ff1 «a jr(ff'a gg) and /(idp a idQ) = idpH Q , hence f : [p,p] a [Q,Q] > [P a Q,P a Qj is an monoid isomorphism. If fQ a po resp. gQ a qQ are dual bases of Ρ and Q then f(fQ a gQ) a (po a q ) is a dual basis of Ρ a Q for jP(fQ Β g0)(P0 a qQ) = fQpQ a Now let p. a f. resp. q. a g. be elements such that '[ρ,ρ] 1 [Q 3 Q] 1 > < >f < > is 1 TNUS P H Now we can define the Brauer group B^(C) of a symmetric monoidal category C . Let A be the(illegitimate) set of isomorphism classes of 1-Azumaya monoids A in C . Then we define an equivalence rela - tion on A by Ä ~ Β iff there exist faithfully projective P,Q ε C such that A a [Ρ,Ρ] = Β a [Q,Q] as monoids. Denote the set of equivalence classes by B^CC) . B^CC) becomes a commutative group in the usual way by [Α] [Β] = [Α Β Β] with unit element [i] and inverse [A°^] for \a] > where A°^ is the 1-Azumaya monoid A with inverse multiplication A a A A a A -f^-> A . Separable monoids Let A be a monoid in C . A is called a separable monoid if the multiplication yu : A a A } A has a splitting σ : A * A a A in ACA such that f-o - idA . Observe that A(X) 3 a ι > a a 1 ε A a A(X) is a splitting for jj. in AC but it is no A-right-morphism. 118 Proposition 6: Let Α ε C be a monoid. Equivalent are a) A is^ separable b) There is an element a H b ε A H A(I) such that i) V c ε A(X): ca B b = a s be ε A a A (X) , ii) ab = 1 ε A(I) . Proof: a) 4 b): Define a Η b: = σ(1). Then 1 = yxc(l) =^t(a Β b) =ab which is condition (ii). Since σ is an A-A-morphism we have ca H b = cc(l) = c(c) = c(l)c = a a be for all a ε A(X). b) 4a): Let σ: A(X) > A a A(X) be defined by o(c) = ca a b . By (i) σ is an A-A-morphism. By (ii) we get yua(c) = yu.(ca a b) = cab = c , hence j±o - idA . Observe that (i) does not depend on a symmetry in C , since aabεABA(I) and I a X = X a I even without a symmetry. The element a a b will be called a Casimir element. Every Casimir element aBbεAaA(I) induces a map Tr: C (Μ,Ν)a f f > (M(X) 5 m ι » af(bm) ε N(X)) ε AC (M,N) for any two objects Μ,Ν ε . In fact for any c ε A(Y) we have ca a b = a a be hence c(af(bm)) = (ca)f(bm) = af(bcm) = af(b(cm)). This map is called the trace map. Since the trace map is a natural transformation, natural in X , Tr: C(M a X,N) > AC(M a Χ,Ν) , M N if we get Tr: [Μ,Ν] >A [ > ] > both objects exist. Since ab = 1 we even get that AC(M,N) > C(M,N) AC(M,N) is the identity on AC(M,N) since Tr(f)(m) = af(bm) = abf(m) = f(m) and hence Tr(f) = f , if f ε AC(M,N) . Similarly A[M,N] ) [Μ,Ν] > A[M,N] is the identity on a[M,N] . If Μ,Ν ε ACA then we clearly get Tr: CA(M,N) > ACA(M,N) and is the identit on C (M N) The C (M N) ACA(M,N) > CA(M,N) A A ' y A A * ' 119 M N same holds for [M,N]a and AL > JA · If a[A,A]a exists then A[A,A]A A[A,A]A —^ [A,A]A is the identity on a[A,A]A . Observe that [A,A]a exists, since [A,A]A = A . Since the last isomorphism is an antiisomorphism of mo [A,A A A[A,A]A > A noids, a ]a is the center of and is a mo noid homomorphism, we get Proposition 7 Ql, Prop. 1.2]: If A is a separable monoid, then the center a[A,A]a (if it exists) is a "direct summand" of A . Let f: A > Β be monoid homomorphism. Ρ ε ßC is called (Β,A)-projective if for each commutative diagram Ν with g,h in gC and k in AC there is a g* εβ0(Ρ,Μ) with hg* = g . The dual notion is that of a (B,A)-injective object [8] Proposition 8: Let A be a separable monoid. Then every A-object is (A,I)-projective and (A,I)-injective. Proof: Let g ε AC(P,N) , h ε AC(M,N) and k ε C(P,M) be given such that hk = g . Then g = Tr(g) = Tr(hk) = h Tr(k) and Tr(k) ε AC(P,M) , so that Ρ is (A,I)-projective.Just by reversing the arrows one can prove that each object in .C is (A,I)-injective. In [8] we prove that in (C,x,E), a monoidal category with the pro duct as tensor-product and E a final object, there are no non-tri - vial finite objects. In Theorem 14 we shall show that [P,P1 is a 120 separable monoid for certain finite objects Ρ ε C . So- this con - struction will not produce examples of separable monoids in (C, χ, E). In fact, there are no non-trivial separable monoids in C at all. Proposition 9 : Let A be a separable monoid in the monoidal category (C, χ, E) . Then A = E as monoids. Proof: Let (a,b) be the Casimir element for A . Then (ca,b) = (a,be) for all c ε A(X) , hence ca = a and b = be . Here we use Α χ A(X) = A(X) χ A(X) and A(E)—>A(X) by the unique mor - phism X ) E . We also have le = c and ab = 1 , hence c = le = abc = ab = 1 for all c ε A(X) , so that A(X) = {1} which proves A = E .Finally observe that E has a unique monoid structure. If A and Β are monoids in C , then Α Η Β is a monoid by (a^ B b1) ' (a2 B b2) = a1a2 Β b1b2 . Proposition 10: Let A and Β be separable monoids. Then Α κ Β is separable. Proof: Let a^ E a2 and b^ B b2 be Casimir elements of A resp. B. Then (a^ Β b^) Β (a2 B b2) is a Casimir element for A Β Β . In fact let χ H y ε A Η B(X) then (χ B y)(a1 H ba) B (a2 B b2)*= (xa1 Η yb^ is (a2 E b2) = (8l1 E ba) H (a2x s b2y) = (a1 Β b^ B (a2 B b2)(x B y) . Furthermore (a. B b.)(a0 E b0) = a.a0 B b.b0 = 1 B 1 . Proposition 11 : Let A be a separable monoid with Casimir element A A a B b . Assume that A[ > 1A exists and that I > A is a mono - morphism. Then A is central if and only if axb ε I(X) for all χ ε A(X). 121 Proof: Since ca H b = a a be for all c ε A(Y) we get A A X for a11 x ε Α χ c(axb) = (axb)c hence axb ε AC > ]A^ ^ ( )· If A is central then axb ε a[A,A] A(X) = I(X) for all χ ε A(X) . Conversely let axb ε I(X) for all χ ε A(X). Let χ ε a[A,Â]A(X) then χ = xab = axb ε I(X) hence a[A,A]A(X) = I(X) . The Brauer group B2(C) A monoid A is called 2-Azumaya if [A,l] and LA3A1 exist and I >A is a monomorphism and if there are elements a H b ε Α Η A(I) and ΟΗάΗθεΑκΑκ A(I) such that i) V Χ ε C V χ ε A(X): xa a b = a a bx , ii) ab = 1 ε A(I) , iii) ac Β dbe = 1 s 1 ε Α Η Α (I) , iv) V Χ ε C V χ ε A(X): axb ε I(X) . Clearly a 2-Azumaya monoid is a central, separable monoid which follows from i), ii), and iv) by Proposition 6 and 11 . We do not know if the existence of c H d Η e with iii) follows from the other conditions. Theorem 12: Let A be a monoid in C . Equivalent are a) A is 2-Azumaya. b) Α ε C is a progenerator and the morphism ψ: Α Β A(X) 3 χ H y ι > ( A( Y) ) 3 ζ ι > xzy ε A(X κ Υ ) ) ε [Α,Α] (Χ) is an isomorphism. c) A is separable and 1-Azumaya. Proof: Let A be 2-Azumaya. Define j\ A H A > [A,l] by Then axzyb ε I(X) by iv) hence f is well-defined. Now 122 f(c a d) a e is a dual basis for A since Now we show that A a [A,I] > I , the morphism induced by the evaluation, is rationally surjective, i.e. that A a [A,I] (I) > 1(1) is surjective. We have to find f^ = (Tr: A » A[A,AJA = I) , the last isomorphism exists in view of Propositions 6 and 11 by the properties i), ii), and iv) of A . Then To show that ψ is an isomorphism we construct the inverse mor - phism [A,A](X)3 σ ι > This morphism is in fact an inverse of ψ since = hence ψ ( xacydb κ e = χ Η acydbe = χ Η lyl = χ Η y . Now assume that b) holds. By Proposition 1 the monoid A is 1-Azumaya. Let f oao a^ be a dual basis for A and sucn that f 1 ε 1 Let a1 a ΐ1 ε Α a [A,I](I) i i = ^ ) · be the element which corresponds to gQ a 1 ε [Α,Ι] B Ad) under the isomorphism A Η A - [Α,Α] = [Α,ι] Η A . Then we have ab = alb = < > xayb = x Assume that c) holds. By Proposition 1 A is faithfully projective and ψ is an isomorphism. Construct a^ a f^ ε A s [Α,Ι](Ι) with We still have to show that b) and c) imply a)» Let f a aQ and 123 2 2 f s a be two copies of the dual basis of A . Let ο ο a1 a f1 ε A a [A,IJ(I) with a s b as above corresponding to gQ and c H d s e := u Η ν κ xy ε Α κ A H A (I) , where υΕν&ΧΗνεΑκΑΗΑΗΑ (I) corresponds to f^ a a2 Η f2 a a^ ε j~A,l] Η Α Η [Α,Ι] Η Ad) under the isomorphisms [Α,Ι] Η Α Η [Α,Ι] S Α = [Α, Α] a LA>A] = A»AaAaA . Then 12 2 1 aczdbe = auzvbxy = 2 2 1 2 2 1 1 2-Azumaya holds, i) and ii) hold by Proposition 1 , iv) by Proposi - tion 11 . Corollary 13: Let A and Β be 2-Azumaya , then A a Β is 2-Azu - maya. Proof: In view of the equivalence of a) and c) in Theorem 12 this follows from Proposition 3 and Proposition 10. Theorem 1*1: Let Ρ ε C be a progenerator. Then [P,P] is 2-Azumaya. Proof: By Proposition 4 we get that Ep>pl ^s l-Äzumaya so that we only have to show that [P,P] is separable. Let f a pQ be a dual < > > basis for Ρ and pa a f ε Ρ a [P,l](I) with P1 f 1 = 1 · Identify [P,P] with [P,l] a Ρ with the multiplication (f a p)(f'a p') = f a f'p'. Then define a a b := (fQ a p1) a (f a ρ ) . For every g a q ε [P,l] a P(X) we have f a p E f K = (g « Ç)( 0 i^ ^ i P0^ (S ® (g a pa) a (f1 a (fQ (fQ a pa) H (f1 a P0)(g « q) so that b) i) of Proposition 6 holds. Furthermore (f0 Β Ρι)(^ . p0) = f0 B It may be interesting to have an explicit description of the eie - ment c κ d H e in the definition of 2-Azumaya for this case [Ρ,Ρ] « Let f* a p^ , i = 1, 2, 3 be copies of the dual basis of Ρ . Then c s d Η e := (f a ρ ) Η (f£ a ρ ) a (f a p^) satisfies condition iii) for 2-Azumaya as is easily checked. To define a Brauer group of 2-Azumaya monoids we need one more lemma. Lemma 15: Let Ρ and Q be progenerators. Then Ρ a Q ±s_ a pro - generator and [Ρ,Ρ] s [QUO] = [P a Q,P a Q] as monoids. Proof: Let p^ a f^ resp. q^ a g^ with (P a Q) a [Ρ a Q,l](I) , where f: [P,lJ a [Q,l] = [P a Q,l] is the isomorphism used in the proof of Proposition 5 . We get < > 1 (p1 a q1 ) (f1 a g1) = Now we can define the Brauer group B^CC), using 2-Azumaya monoids, in the same way as B^(C) . Since each 2-Azumaya monoid is 1-Azumaya and since each progenerator is faithfully projective we get a group homomorphism ξ: B2(C) > ^l^^ * Since the notions of progenerator and faithfully projective coincide, if I ε C is projective, the notions of 1-Azumaya and 2-Azumaya coincide by Theorem 12, b) and Pro- 125 position 1 . So does the equivalence relation used in the construction of the two Brauer groups and we get Theorem 16: The group homomorphism ξ: B2(C) } B^(C) _is_ the identity in case I ε C _is projective. Splitting Azumaya monoids by monoidal functors. Now we want to discuss the behaviour of the Brauer groups under a monoidal functor. Let C and V be symmetric monoidal categories and F: C > V be a covariant functor. Denote the tensor products and the associativity, the symmetry and unity isomorphisms in C and V by the same signs Β, α, y , λ, and j . Assume that there are na - turai transformations 6: FX Β FY > F(X B Y) ζ: J > Fl such that the following diagrams commute 1 B ζ FX Β FI < — FX E J <5 S F( } F(X Η I) — ? > FX ζ H 1 FI s FX < — J Η FX 6 λ v Fm ^ F(I Β X) > FX 1 6 FX Β (FY a FZ) ® > FX H F(Υ Η Ζ) F(X Η (Υ Η Ζ)) α F(a) ψ ^ 6 g (FX a FY) κ FZ S F(X Β Υ) Η FZ -^4 F((X Β Υ) Β Ζ) . If C and V are symmetric we require in addition the commutativity of 126 FX a FY FY a FX δ 6 F(X H Y) F(N > F(Y H X) . Such a triple (F, δ, ζ) will be called a weakly monoidal functor. Let π: Χ a [Χ,Υ] >Y and τ: Y » [Χ,Χ a Y] be front and back adjunction for the adjoint pair of functors X a- and [X*-] 5 if [xs~] exists. Again we use the same notation in both categories C and V . Let χ: C(X a Y,Z) = C(Y, [x,z]) and ω: C(Y, [X,z]) = C(X a Y,Z) be the corresponding adjointness isomor - phisms in C resp. also in 0 .It is an easy exercise in diagram chasing for adjoint functors to show that there is a natural trans - formation Φ : F[X,Y] 9 [FX,FY] whenever [Χ,Υ] and [FX,FY] exist, just take Φ = χ(Ρ(π)6) . Furthermore the diagrams FX a F [Χ,Υ] —>F(X a [X,Y] ) FY Ρ(Τ) > F[X,X a Y] 1 - Φ I F(TT) |Τ l Φ FX a [FX,FY] ^ FY EFX>FX " FY^ [FX,F(X a Y)] and J [FX,FX] Jc u Fi ISJâ^ F [Χ,Χ] commute. Here i: I > Cp>pl ^s x(^) where j> : X a I > X and j is defined analogously in Ό . Omitting special arrows for the associativity α we get the commutative diagram on the next page. If we abbreviate [ΐ,π a i] τ by j> : [Χ,Υ] a Ζ > [Χ,Υ a z} then the following diagram (the outer frame of the given diagram) commutes F [Χ,Υ] a ΓζΛ F([X,Y] a Z) IUI- F([x,Y a z] ) 1 Φ [FX,FY] a FZ [FX, FY a Fz] l1'^ > [FX,F(Y a Z)] . F T F([X,Y] « Ζ) til) > F[X,X . [χ,Υ] ι Ζ] ^> ' ^ ? F [Χ,Y . Ζ] ^ Γ Ί V F(lT 88 [FX,FX . F([X,Y] is Ζ)] IM-I > [FX,F(X is [Χ,Υ] a Z)] ^> [FX,F(Y Β Ζ )] A [1,1 is δ] F[X,Y] IS FZ -1+ [FX,FX is F[X,Y] IS FZ] ί1*6 8 ^ > [FX,F(X IS [Χ,Υ] ) is FZj DM] Φ Β 1 [1,1 Β Φ Β 1} [ΐ,Ρ(π) Β 1] [FX,FY] Β FZ -I» [FX, FX Β [FX, FY] Β FZ] g ^ > [FX, FY Β FZ] 128 Theorem 17 : Let F: C * V be a weakly monoidal functor. Assume that ζ : J FI is an isomorphism and that ζ: F[P,l] B FP > F([P,l] Β Ρ) is an isomorphism for all finite ob - jects Ρε C , If Ρ is finite in C and if [FP,-] exists in V then FP is finite in V . Proof: Since finiteness is equivalent to the fact that i: I > [Ρ,Ρ] can be factored through [Ρ,ΐ] Β Ρ the following commutative diagram shows that j: J } [FP,FP] can be factored through [FP,J] Β FP [FP,j] Β FP [FP,J Β FP] thus FP is finite. Corollary 18 : Under the assumption of Theorem 17 is the morphism Φ : F[P,X] ^ [FPjFX] an isomorphism for all Χ ε C and all finite Ρ ε C . Proof: Let f Β pQ: I > [Ρ,ΐ] Β Ρ be the dual basis for Ρ and fo Β pQ: J ϊ [FP,J] ® FP be the dual basis for FP . Define Ψ : [FP,FX] > F[P>X] to be 129 [FP,FX] - FI B [FP,FX] * F([P,l] ι P) a [FP,FX] ^-1^ 1 π F[P,I] Β FP κ [FP,FXj * ) F[P,l] a FX —> F]_P3X] . Omitting some of the obvious isomorphisms we get a commutative dia - 1 Η Τ [FP,FXJ > F[P,l] • FX —) F[P,X] 1 g 71 [FP,FI] a FP a [FP3Fx] ) [FP,Fl] a FX = [FP,FX] where the left triangle commutes by the construction of f a pQ in Theorem 17 and right square commutes in the same way as the middle of the diagram in the proof of Theorem 17 does. If we look at the lower part of our diagram we see that the morphism FP Fx is the [FP,FXj » L > J identity since ?0 F[P,x] > F[P,I] a FP a F[p,x] 1 2 6 > F[P,fj a F(P a [P,x]) |φ I 1 a 1 Β Φ J/lH F^ 1 π [FP,FX] > F[P,I] a FP a [FP3FX] ® ) F[P,l[ κ FX > F[P,X] shows ΨΦ = id . Corollary 19 : Under the assumptions of Theorem 17 if Ρ is a pro - generator then FP is a progenerato^. If F preserves difference cokernels and Ρ is faithfully projective then FP i_s faithfully projective. Proof: Let Ρ be finite. Ρ is a progenerator iff there is a mor - phism f: I ) Pa [P,l] such that 130 Ρ Β [Ρ,ΐ] commutes. Now the diagram F(f)C F(p Β Ιΐ,τ}) IR (1 Β [ι,ζ-1] )(1 Β Φ)6_1 FP Β [FP.jQ > commutes hence FP is a progenerator. In the case of a faithfully projective Ρ we have to replace Ρ Η [P,l] by Ρ Η [Ρ,I] and FP Β [FP,,!] by [p>p] P J FP Β LF 5 1 = F(P ss [P,l]) . The last isomorphism is a conse [FP,FP] [Ρ,Ρ] quence of the fact that F preserves difference cokernels. Theorem 20: Let F: C > Ρ be a weakly monoidal functor such that ζ: J > FI is an isomorphism, δ: FX Β FP > F(X s Ρ) is_ an iso - morphism for all Χ ε C and for all finite Ρ ε C , [FP,-Q exists for all finite Ρ ε C and F preserves difference cokernels. Then F induces homomorphisms of Brauer groups B.(F): B.(C) > B.(P) BTCfn 1 1 1 for i = 1, 2 such that B2(C) —-—> B2(P) B1(C) —i—> B1(P) commutes. Proof: Let A be a monoid in C . Then FA is a monoid in V with the multiplication FA E FA > F(A B A) > FA and unit ζ V( ì J » FI ry2 > FA . If A is i-Azumaya, i = 1, 2, then FA is faithfully projective resp. a progenerator by Corollary 19, Proposi • tion 1 and Theorem 12. So we only have to show that ψ: FA κ FA > [FA,FA2 is an isomorphism, ψ is induced by 131 Τ: A H A H A(X) a a H b H c ι > bac ε A(X) , so that ψ = χ(Τ) where Χ: C(X Β Υ,Ζ) = C(Y,[x,z]) . Now the diagram FA Β FA X(T) > [FA,FA] δ Φ F( (T)) F(A Β A) X F[A,A] commutes, since FA Β (FA Β FA) FA As 1 Β δ F(Τ) FA Β F (Α Β Α) —F( Α Β (Α Β Α)) commutes so that by applying χ we get Φ 0 F(x(T))«6 = χ(Ρ(Τ)«δ)οδ = [l,F(T)]o χ(δ)οδ = χ(Τ) . The first identity results from the commutativity of C(X Β Y,Ζ) P(F(X Β Y) , FZ) Ρ(6^1) ) P(FX B^FY,FZ) Ρ(1?Φ) C(Y, [Χ,Ζ]) -Ì-> P(FY,F[X,Z]) ) fl(FY,[FX,FZ]) . Now Φ: F [A,A] > [FA,FA] is an isomorphism by Corollary 18 and F(x(T)) = Ρ(ψ) is an isomorphism since A is Azumaya. Since δ is an isomorphism, too, we get that ψ= χ(Τ): FA Β FA ) -[FA,FA] is an isomorphism. If Ρ ε C is faithfully projective or a progenerator in C then as above FP is faithfully projective or a progenerator in V and F[P,P] = [FP,FP] as monoids using the first commutative diagram we proved for Φ . Thus if A and Β are i-Azumaya, then FA and FB are i-Azu - maya. If A and Β are equivalent w.r.t. B^(C) , then so are FA and FB . Finally we have F(ABB)=FABFB and FI = J so that F induces homomorphisms B^(F): 8^(C) > Bi^ such that the dia - gram in the theorem commutes. If F: C > V is a functor satisfying the conditions of Theo - rem 20 then we define the kernel of as B^(C,F) so that we get exact sequences 132 Ο > B.(C,F) » BI(C) > Β.(Ρ) for i = t, 2 . BI(C,F) containes those elements [A] of B^(C) with [FA] = [[P,P]J for some Ρ ε V which is faithfully projective resp. a progenerator. These i-Azumaya monoids A are called F-split. From Theorem 20 follows immediatly a homomorphism ξ: B2(C,F) > B1(C,F) . If C is a symmetric monoidal closed category with difference kernels and difference kokernels and Κ ε C is commutative monoid, then KC is again a symmetric monoidal closed category with a^ as tensor product and Κ as basic object. Then the functor C 3 X I—^ Κ H Χ ε „C has all properties required in Theorem 20 hence there are homomorphisms B^(C) —^ ^i^K^ with kernels B^(K/C) . 133 References : [l] Auslander, M. and Goldmann, 0.: The Brauer group of a commutative ring, Trans.Amer.Math.Soc. 97 (I960), 367 - 409. [2] Long, F.W.: The Brauer group of dimodule algebras, J. of Algebra 30 (1974), 559 - 601. [3] MacLane, S.: Categories for the working mathematician, Graduate Texts in Mathematics. Springer New York - Heidelberg - Berlin 1971. [4] Orzech, M. and Small, Ch.: The Brauer group of commutative rings. Leisure notes in Pure and Applied Mathematics. Marcel Dekker New York 1975. [5] Pareigis, B.: Non-additive ring and module theory I: General theory of monoids. To appear in: Publicationes Mathematicae Debrecen. [6] Pareigis, B.: Non-additive ring and module theory II: C-catego - ries, C-functors and C-morphisms. To appear in: Publicationes Mathematicae Debrecen. [7] Pareigis, B.: Non-additive ring and module theory III: Morita theorems over monoidal categories. To appear in: Publicationes Mathematicae Debrecen. [8] Pareigis, B.: Non-additive ring and module theory V: Projective and flat objects. To appear in: Algebra-Berichte. [9] Fisher-Palmquist, J.: The Brauer group of a closed category, Proc.Amer.Math.Soc. 50 (1975), 6l - 67. gg' =
goqo = ρ a q . Hence Ρ a Q is finite.
fQP1) a (f1 a pQ) =
fQpQ) = (g a p1) a (f1 a q) = 124