A Luna étale slice theorem for algebraic stacks. (English) Zbl 1461.14017

Alper, Jarod; Hall, Jack; Rydh, David

Ann. Math. (2) 191, No. 3, 675-738 (2020).

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The paper contains several foundational results on algebraic stacks \(\mathscr{X}\) that are locally of finite type over algebraically closed fields \(k\), and have affine stabilizers. The main result is the following: Suppose \(x\in \mathscr{X}(k)\) is a rational point, \(G_x\) the stabilizer group scheme, and \(H\) be a subgroup scheme that is linearly reductive, with \(G_x/H\) smooth. Then there is an smooth morphism \[ f:([\operatorname{Spec}(A)/H], w)\longrightarrow (\mathscr{X},x) \] for some affine scheme \(\operatorname{Spec}(A)\) with \(H\)-action and rational fixed point \(w\), such that the stack of \(H\)-torsors \(BH\) is equivalent to the pullback \(f^{-1}(BG_x)\). Moreover, one may assume that \(f\) is étale if \(G_x/H\) is étale, and affine if \(\mathscr{X}\) has affine diagonal. The result confirms that the quotient stacks \([\operatorname{Spec}(A)/H]\) are indeed the basic building blocks for algebraic stacks of the above type with linearly reductive stabilizers.
A more refined version holds if additionally \(x\in\mathscr{X}\) is smooth and the whole stabilizer \(G_x\) is linearly reductive: Then there is a commutative diagram \[ \begin{tikzcd} ([N_x/G],0) \longleftarrow \arrow[d] & ([\operatorname{Spec}(A)/G_x]),w) \arrow[r, "f"] & (\mathscr{X},x) \\ (N_x/\!\!/G_x,0) \arrow[r] & (U,u) \end{tikzcd} \] where \(f\) is étale and induces an isomorphism of stabilizers, the square is cartesian, and the lower arrow is an affine étale morphism from an affine scheme \(U\) to the the GIT quotient \( N_x/\!\!/G_x\). Here \(N_x=(\mathscr{I}/\mathscr{I}^2)^\vee\) denotes the normal space for \(x\in\mathscr{X}\) viewed as a \(G_x\)-representation.
The authors give many applications, both of local and global nature. To mention a few, they establish the existence of equivariant miniversal deformation spaces for singular curves, compact generation of derived categories of certain algebraic stacks, and Białynicki-Birula decompositions for Deligne-Mumford stacks with \(\mathbb{G}_m\)-actions.
A key ingredient for the proof of the main results is a theory of coherent completeness for algebraic stacks. If \(A\) is a complete local noetherian ring, the finitely generated \(A\)-modules can be seen as compatible systems of finitely generated modules over \(A/\mathfrak{m}_A^{n+1}\). The authors shows that under suitable hypothesis a similar equivalence of categories \[ \operatorname{Coh}(\mathscr{X}) \longrightarrow \varprojlim \operatorname{Coh}(\mathscr{X}^{[n]}_x) \] holds for algebraic stacks \(\mathscr{X}=[\operatorname{Spec}(A)/G]\). Another important ingredient are results on Artin Algebraization in an equivariant setting, which are developed in the appendix.

Reviewer: Stefan Schröer (Düsseldorf)

MSC:

14D23

Stacks and moduli problems

Keywords:

algebraic stacks; quotients; moduli spaces; equivariant geometry