Quotient Stacks - Key Properties

Quotient stacks inherit geometric properties from both the scheme being quotiented and the group acting on it. Understanding these properties is crucial for applications.

TheoremSmoothness of Quotient Stacks

Let $G$ act on a smooth scheme $X$ . Then $[X/G]$ is a smooth stack if and only if the action is smooth (the action morphism $G \times X \to X$ is smooth). When $G$ is smooth and the action is free, $[X/G]$ is automatically smooth.

More generally, the smoothness locus of $[X/G]$ consists of points where the stabilizer acts smoothly.

DefinitionGood and Adequate Moduli Spaces

A morphism $\pi: \mathcal{X} \to X$ from a stack to an algebraic space is a good moduli space if:

$\pi_*\mathcal{O}_\mathcal{X} = \mathcal{O}_X$
$\pi_*: \textbf{QCoh}(\mathcal{X}) \to \textbf{QCoh}(X)$ is exact
Formation of $\pi_*$ commutes with flat base change

For quotient stacks $[X/G]$ , good moduli spaces are closely related to GIT quotients.

TheoremCohomology of Quotient Stacks

For $[X/G]$ with $G$ acting properly and $\mathcal{F}$ a $G$ -equivariant sheaf: $H^*([X/G], \mathcal{F}) = H^*_G(X, \mathcal{F})$ the $G$ -equivariant cohomology. When $G$ is finite, this equals: $H^*([X/G], \mathcal{F}) = H^*(X, \mathcal{F})^G$ the $G$ -invariant part of ordinary cohomology.

ExampleChow Rings of Quotient Stacks

The Chow ring of a quotient stack can be computed as: $A^*([X/G]) = A^*_G(X)$ the equivariant Chow ring. For finite $G$ : $A^*([X/G]) = (A^*(X) \otimes R(G))^G$ where $R(G)$ is the representation ring.

TheoremSeparated Quotient Stacks

$[X/G]$ is separated if and only if the action is proper: the action morphism $G \times X \to X \times X$ given by $(g,x) \mapsto (x, g \cdot x)$ is proper. For affine groups, this is automatic.

DefinitionLinearization

A $G$ -linearization of a line bundle $L$ on $X$ is an isomorphism $\phi: \sigma^*L \xrightarrow{\sim} \text{pr}_2^*L$ satisfying compatibility conditions. Linearized line bundles descend to line bundles on $[X/G]$ .

The assignment $(L, \phi) \mapsto L^G$ gives an equivalence: $\text{Pic}_G(X) \xrightarrow{\sim} \text{Pic}([X/G])$

TheoremBase Change for Quotient Stacks

Given $H \subset G$ a subgroup and $H$ acting on $Y$ , if $f: Y \to X$ is $H$ -equivariant (where $H$ acts on $X$ via the inclusion), then: $[Y/H] \to [X/G]$ and fiber products $[Y/H] \times_{[X/G]} T$ can be computed as quotient stacks of fiber products at the scheme level.

Remark

Many classical results about group actions on schemes (Luna's slice theorem, Hilbert-Mumford criterion, etc.) have stacky analogs. The quotient stack framework makes these results more natural and often easier to prove.

These properties make quotient stacks computable and allow explicit calculations in many examples of interest in geometry and representation theory.