Bogomolov, F. A.; Landia, A. N. 2-cocycles and Azumaya algebras under birational transformations of algebraic schemes. (English) Zbl 0729.14014 Compos. Math. 76, No. 1-2, 1-5 (1990). 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) Cited in 1 Document MSC: 14F99 (Co)homology theory in algebraic geometry 14C25 Algebraic cycles 14F22 Brauer groups of schemes 57R95 Realizing cycles by submanifolds Keywords:cohomology class as Azumaya algebra; second cohomology; Brauer group PDF BibTeX XML Cite \textit{F. A. Bogomolov} and \textit{A. N. Landia}, Compos. Math. 76, No. 1--2, 1--5 (1990; Zbl 0729.14014) Full Text: Numdam EuDML OpenURL References: [1] Bogomolov, F.A. , Brauer group of quotients by linear representations . Izv. Akad. Nauk. USSR, Ser. Mat. 51 (1987) 485-516. · Zbl 0641.14005 [2] Landia, A.N. , Brauer group of projective models of quotients by finite groups . Dep. in GRUZNIITI 25.12.1987, no. 373-r87. [3] Moishezon, B.G. , An algebraic analog of compact complex spaces with sufficiently large field of meromorphic functions I . Izv. Akad. Nauk. USSR, Ser. Mat. 33 (1969) 174-238. · Zbl 0197.17502 [4] Hartshorne, R. , Algebraic Geometry . Graduate Texts in Math . 52, Springer Verlag, Berlin etc. 1977. · Zbl 0367.14001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.