A cohomological interpretation of Brauer groups of rings

Abstract

Quillen's proof of the Serre conjecture introduced a new tool for passing from local to global results on affine schemes. We use this to prove the theorem below characterizing the image of the injection i: Br{X) —• H2(Xct, Gm) when X = Spec A, is a regular scheme. A result of M. Artin then allows us to conclude that Br(X) = H%Xet, Gm) if X = Spec A is a smooth, affine scheme over a field. For such rings, this proves the Auslander Goldman conjecture [2], Br{Λ) — f)Br(A p), peP(A), the set of height one primes of A. We begin with following theorem. THEOREM. Let X = Spec A be a regular scheme. If ce H2(Xet, Gm) and cy = i([^y) in H 2(Sipec(Amy)et, Gm) for all closed points y e X, then c Proof. If /eA, let cf denote the restriction of c to H2(Sj?ec(Af).t, GJ . Let S = {f e A\cf = ί([Λ]) for some Azumaya algebra A over Af}. We will show that S is an ideal. Then S = A since the hypothesis on c prevents S from being contained in any maximal ideal of A.