## 2-cocycles and Azumaya algebras under birational transformations of algebraic schemes.(English)Zbl 0729.14014

The authors establish the following theorem: Let X be a Noetherian scheme, and let $$\gamma$$ be an element of $$H^ 2(X,{\mathcal O}^*_ X)$$. Then there is a scheme Y and a proper birational morphism $$\alpha: Y\to X$$ such that the cohomology class $$\alpha^*(\gamma)$$ is represented by an Azumaya algebra on Y. - The proof is by a relatively brief inductive argument, which the authors describe as a simple version of a proof due to O. Gabber.
When second cohomology is a birational invariant (for example, for smooth projective varieties), the theorem comes very close to identifying second cohomology and the Brauer group, and in fact does so when second cohomology is finite. As the authors remark, both these conditions obtain for a nonsingular projective model X of V/G where G is a finite group and V is a faithful complex representation of G (using the result of the first named author that in this case $$H^ 2(X,{\mathcal O}^*)$$ is isomorphic to the finite group $$H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}}))$$.
Reviewer: A.R.Magid (Norman)

### MSC:

 14F99 (Co)homology theory in algebraic geometry 14C25 Algebraic cycles 14F22 Brauer groups of schemes 57R95 Realizing cycles by submanifolds
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### References:

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