Quotient Stacks - Core Definitions

Quotient stacks $[X/G]$ are the fundamental building blocks of stack theory, providing a systematic way to construct stacks from group actions on schemes.

DefinitionQuotient Stack

Let $G$ be an algebraic group acting on a scheme $X$ over $S$ . The quotient stack $[X/G]$ is the stack defined by: $[X/G](T) = \{P \to T \text{ a principal $G$-bundle}, f: P \to X \text{ a $G$-equivariant morphism}\}$

Morphisms in $[X/G](T)$ are isomorphisms of $G$ -bundles compatible with the maps to $X$ . This construction captures the geometry of the action while retaining automorphism information.

The natural projection $X \to [X/G]$ sending $x$ to the trivial bundle with map to $x$ is a smooth atlas when $G$ acts freely, and more generally provides the canonical presentation of the stack.

ExampleThe Classifying Stack

When $G$ acts trivially on a point, $[\text{pt}/G] = \mathcal{B}G$ is the classifying stack. It has a single isomorphism class of objects (the trivial bundle) with automorphism group $G$ . Every quotient stack $[X/G]$ maps naturally to $\mathcal{B}G$ (forgetting the map to $X$ ).

TheoremQuotient Stacks Are Algebraic

If $X$ is a scheme and $G$ is an algebraic group over $S$ with $X \to S$ and $G \to S$ of finite type, then $[X/G]$ is an algebraic stack. Moreover:

The natural morphism $X \to [X/G]$ is smooth and surjective
If $G$ acts freely, then $X \to [X/G]$ is a $G$ -torsor
$[X/G]$ is Deligne-Mumford if and only if $G$ is finite and acts with finite stabilizers

DefinitionEquivariant Sheaves

A $G$ -equivariant quasi-coherent sheaf on $X$ consists of a quasi-coherent sheaf $\mathcal{F}$ together with an isomorphism $\alpha: \sigma^*\mathcal{F} \xrightarrow{\sim} \text{pr}_2^*\mathcal{F}$ on $G \times X$ satisfying a cocycle condition.

The category of $G$ -equivariant sheaves is equivalent to $\textbf{QCoh}([X/G])$ : $\textbf{QCoh}_G(X) \simeq \textbf{QCoh}([X/G])$

ExampleProjective Space as Quotient Stack

Projective space can be written as: $\mathbb{P}^n = [(\mathbb{A}^{n+1} \setminus \{0\})/\mathbb{G}_m]$ where $\mathbb{G}_m$ acts by scaling. The coarse moduli space of this stack is the usual projective space $\mathbb{P}^n$ . Line bundles on $\mathbb{P}^n$ correspond to $\mathbb{G}_m$ -equivariant sheaves on $\mathbb{A}^{n+1} \setminus \{0\}$ , i.e., graded modules.

Remark

The quotient stack $[X/G]$ should be distinguished from the geometric quotient $X/G$ (when it exists). The stack retains all automorphism information, while $X/G$ forgets it. When $X/G$ exists as a scheme or algebraic space, it is typically the coarse moduli space of $[X/G]$ .

DefinitionStabilizer Groups

For a quotient stack $[X/G]$ and a geometric point $\bar{x} \in [X/G]$ , the stabilizer group is: $\text{Stab}(\bar{x}) = \{g \in G : g \cdot \bar{x} = \bar{x}\} \subset G$

The inertia stack $I_{[X/G]}$ consists of pairs $(x, g)$ where $g \in \text{Stab}(x)$ . The projection $I_{[X/G]} \to [X/G]$ is finite if and only if all stabilizers are finite (i.e., $[X/G]$ is Deligne-Mumford).

Quotient stacks are ubiquitous in algebraic geometry, appearing whenever symmetries are present. They provide the natural setting for geometric invariant theory and moduli problems.