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

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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.

### MSC:

 14D23 Stacks and moduli problems

### Keywords:

algebraic stacks; quotients; moduli spaces; equivariant geometry
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