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The Brauer group of quotient spaces by linear group actions. (English. Russian original) Zbl 0679.14025

Math. USSR, Izv. 30, No. 3, 455-485 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 485-516 (1987).
The author studies an important birational invariant of the fields of invariants of an algebraic group. For every field \(K\) of algebraic functions let \(Br_ v(K)\) denote the unramified Brauer group of \(K\). It is defined as the subgroup of the Brauer group of K equal to the intersection of the kernels of the localization homomorphisms corresponding to all discrete valuations of \(K\). More geometrically, it can be defined as the Brauer-Grothendieck group of a nonsingular complete model of \(K\). For every algebraic group \(G\) acting on a nonsingular complete algebraic variety \(X\), let \(Br_ v(X/G)\) denote the unramified Brauer group of the field of invariants \({\mathbb{C}}(X)^ G\). Assume that \(G\) acts freely on an open subset \(U\) of \(X\) and the geometric quotient \(U/G\) exists. Then there is a natural map \(r: H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}})\to Br(U/G).\)
The first result characterizes the maximal subgroup \(B^ G_ X\) of \(H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}})\) the image of which is contained in \(Br_ v(X/G)\). An element \(\gamma\) belongs to \(B^ G_ X\) if and only if its restriction \(\gamma_ A\) to any abelian rank 2 subgroup \(A\) of \(G\) belongs to \(B^ A_ X\). The latter condition is satisfied if and only if \(\Gamma_ A\) restricts trivially to any subgroup of \(A\) which acts cyclically on some smooth subvariety of \(X^.\), where \(X^.\) is \(A\)-equivariantly birationally isomorphic to \(X\).
The next result gives an important application of this theorem. Let \(H\) and \(K\) be two subgroups of a simply connected algebraic group \(G\), and \(H*K\) be the factorgroup of \(H\times K\) by the kernel of the left-right action of \(H\times K\) on \(G\). If \(H*K\) acts freely on an open subset (generically freely) of \(G\) then \(Br_ v(H\setminus G/K)\cong Br_ v(V/H*K)\), where \(V\) is any linear generically free representation of \(H*K\). The latter group was studied in an earlier paper of the author [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 485–516 (1987; Zbl 0641.14005)]. Finally the author relates the group \(Br_ v(K)\) to the Galois group \(\text{Gal}(K)\) of a field of algebraic functions \(K\). He proves that \(Br_ v(K)\) coincides with the subgroup of \(H^ 2(\text{Gal}(K),{\mathbb{Q}}/{\mathbb{Z}})\) which consists of elements which restrict trivially to any abelian rank 2 subgroup of \(\text{Gal}(K)\).
Reviewer: I. V. Dolgachev

MSC:

14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
14F22 Brauer groups of schemes
14E05 Rational and birational maps

Citations:

Zbl 0641.14005
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