An algebraic stack $\mathcal{X}$ is a global quotient $[X/G]$ for some scheme $X$ and algebraic group $G$ if and only if:

1. $\mathcal{X}$ admits a flat atlas $p: U \to \mathcal{X}$ such that the groupoid $U \times_\mathcal{X} U \rightrightarrows U$ is flat
2. The inertia stack $I_\mathcal{X}$ is affine over $\mathcal{X}$
3. $\mathcal{X}$ satisfies certain technical conditions on stabilizers

For Deligne-Mumford stacks with finite stabilizers, every such stack is étale-locally a quotient by finite groups.

For $[X/G]$, morphisms $T \to [X/G]$ from a scheme $T$ correspond to:

• Principal $G$-bundles $P \to T$
• $G$-equivariant morphisms $P \to X$

This universal property characterizes quotient stacks and makes them functorial in both $X$ and $G$.

Let $G$ act on $X$ with an ample $G$-linearized line bundle $L$. Then:

1. The open substack $[X^{ss}(L)/G] \subset [X/G]$ of semi-stable points admits a good moduli space
2. This good moduli space is the GIT quotient $X^{ss}(L)//G$
3. The morphism $[X^{ss}(L)/G] \to X^{ss}(L)//G$ is proper and satisfies $\pi_*\mathcal{O} = \mathcal{O}$

This theorem unifies classical GIT with modern stack theory.

For a proper Deligne-Mumford stack $\mathcal{X}$ that is a global quotient $[X/G]$ with $X$ proper and $G$ finite, the cohomology satisfies: $H^*(\mathcal{X}, \mathbb{Q}) = H^*(X, \mathbb{Q})^G$ and there is a rational Chow ring: $A^*(\mathcal{X})_\mathbb{Q} = A^*(X)_\mathbb{Q}^G$

The rational coefficients are essential due to orbifold phenomena.

Every smooth Deligne-Mumford stack $\mathcal{X}$ with finite stabilizers is étale-locally a quotient $[U/\Gamma]$ where $U$ is a smooth scheme and $\Gamma$ is a finite group. Moreover, $\mathcal{X}$ admits a stratification by smooth global quotient stacks.

Let $G$ act on a smooth scheme $X$ and $x \in X$ with closed orbit. Then étale-locally near $x$, the quotient stack $[X/G]$ is isomorphic to $[V/H]$ where:

• $V$ is a linear representation of $H = \text{Stab}(x)$
• The isomorphism is compatible with the $H$-action

This provides local models for singularities of quotient stacks.

For a normal variety $X$ with an action of a torus $T$, every point has a $T$-invariant affine open neighborhood. This implies that $[X/T]$ can be covered by quotients of affine schemes, simplifying many constructions.

For a quotient stack $[X/G]$ with $G$ acting properly, the K-theory satisfies: $K_0([X/G]) = K^G_0(X)$ the $G$-equivariant K-theory of $X$. There is a Chern character: $\text{ch}: K_0([X/G]) \to A^*([X/G])_\mathbb{Q}$ satisfying Grothendieck-Riemann-Roch.