Brauer groups and quotient stacks.

*(English)*Zbl 1036.14001Since the quotient of a scheme by a group need not exist as a scheme, it is often helpful to consider quotients as stacks, and a natural question is to determine which algebraic stacks are quotient stacks.

The authors of the paper under review give some partial answers to this question:

(i) All orbifolds are quotients stacks;

(ii) All regular Deligne-Mumford stacks of dimension \(\leq 2\) are quotient stacks; and

(iii) There exists a Deligne-Mumford stack, that is normal and of finite type over the complex numbers (but singular and nonseparated) which is not a quotient stack.

The above question is related to an old question of whether, for a scheme \(X\), the natural map from the Brauer group to the cohomological Brauer group, the torsion group of \(H^2_{\text{ét}}(X,\mathbb{G}_m)\), is surjective. The main result of the paper is that a stack is a quotient stack if and only if a certain class in the cohomological Brauer group associated with it lies in the image of the map from the Brauer group. So (iii) yields an example of the nonsurjectivity of the Brauer map for a finite-type normal, but nonseparated, scheme. A different proof of the last result has also been given in S. Schröer’s paper [J. Algebra 262, 210–225 (2003; Zbl 1036.14009)].

The authors of the paper under review give some partial answers to this question:

(i) All orbifolds are quotients stacks;

(ii) All regular Deligne-Mumford stacks of dimension \(\leq 2\) are quotient stacks; and

(iii) There exists a Deligne-Mumford stack, that is normal and of finite type over the complex numbers (but singular and nonseparated) which is not a quotient stack.

The above question is related to an old question of whether, for a scheme \(X\), the natural map from the Brauer group to the cohomological Brauer group, the torsion group of \(H^2_{\text{ét}}(X,\mathbb{G}_m)\), is surjective. The main result of the paper is that a stack is a quotient stack if and only if a certain class in the cohomological Brauer group associated with it lies in the image of the map from the Brauer group. So (iii) yields an example of the nonsurjectivity of the Brauer map for a finite-type normal, but nonseparated, scheme. A different proof of the last result has also been given in S. Schröer’s paper [J. Algebra 262, 210–225 (2003; Zbl 1036.14009)].

Reviewer: Jannis A. Antoniadis (Iraklion)