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Additional resources for Brauer Groups: Proceedings of the Conference Held at Evanston, October 11–15, 1975

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The Amitsur cohomology set is HI(s/R, ASM) = ZI(s/R, A_sM)/A_s(M). Then HI(s/R, ASM) is a functor on D_* as follows. For <~, A>: M + N and a ( ZI(s/R, A_sM) let <~, A>a = s0 9 Aa "Ol-I where Aa = A | a E A_s2(AMl , AMo). ] A : U by Z2(S/R,UsM) : {u ~ U_s3(MIo, MOO) : UoU2 = UlU3}. U_s2(M) operates via v(u) = VoV2Vl-lu. Then 54 H2(S/R, U_sM) = Z2(S/R, UsM)/U_s2(M) is a constant (abelian group) functor on D_*. In our situation H0 seemsthe most d i f f i c u l t cohomology set to define. shall only define i t for A = PIC.

E) (a,b) @ ( a , b ' ) ~ (a,b+b') @ (O,b) similar to so (a,b) @ ( a , b ' ) is (a,b+b'). Call Br(R)p power exponent. the subgroup of Br(R) We study the subgroup of all elements of p- Br(R)p by first studying its subgroup consisting of all elements of exponent this last subgrcup is generated by the p. (a,b)R'S. We claim that This is a conse- quence of the following. 6: If R[a~/p-'''''~n~ each Ai A is Azumaya over then is split by A R and split by is similar to A I @ A 2 @ ... | A n 8omething stronger is, in fact, true.

3) P S Mn(S @ S) an ~tale covering of R is a free module, that is, we can consider the isomorphism phism of Suppose a is such that P = S n. 5) as the inner autormor- determined by a. By [KO] p. 38 identified with the following subring of {x E Mn(S) I ,(l@x) = x@l]. A R. Mn(S), Choose an extension A can be A = f: > S' . S f induces a map f: Mn(S @ S) > Mn(S t @ S ') and call u f = f(a). u f induces, by conjugation, an automorphism ,f of Mn(S' | S' ). Since $2 = ~3~1 ' ~2f = $3f,1 f. fully flat descent, an algebra Af R' is Azumaya over > Mn(S') clearly tf If ~ ,f determines, by faith- A f = Ix E Mn(S') I ,f(l@x) = x@l}.

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