# 2024-07-23-14-30-14

## Review

:::{.remark}
Recall from yesterday that an algebraic stack is a stack $\cX$ over the big etale site $\Sch_\Et$ such that there exists a scheme $U$ and a representable, smooth, surjective map $U\to \cX$, i.e. a smooth presentation.
A DM stack replaces smooth with étale, and an algebraic space is an algebraic stack where all stabilizers are trivial.
Fact: the diagonal is representable.

\[\begin{tikzcd}
	\cX && {\cX \times \cX} \\
	\\
	{\Isom(a, b)} && T
	\arrow[from=1-1, to=1-3]
	\arrow[from=3-1, to=1-1]
	\arrow[from=3-1, to=3-3]
	\arrow["{a,b}"', from=3-3, to=1-3]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGNYIl0sWzIsMCwiXFxjWCBcXHRpbWVzIFxcY1giXSxbMCwyLCJcXElzb20oYSwgYikiXSxbMiwyLCJUIl0sWzMsMSwiYSxiIiwyXSxbMiwzXSxbMiwwXSxbMCwxXV0=)

The stabilizer of $x: \spec k \to \cX$ is $G_x = \Isom_k(x, x) \da \Aut_k(x)$.

:::

:::{.proposition}
If $\cX$ is a Noetherian algebraic stack and $x\in \abs{\cX}$ is finite type, then $\exists \cG_x \injects \cX$ a locally closed substack with $\abs{\cG_x} = \ts{x}$ and $\cG_x$ representable.
:::

:::{.remark}
See *residual gerbes* $\cG_x$.
Letting $[C]\in \Mg$ be a curve class, the residual gerbe is the classifying stack $\cG_{[C]} = \B \Aut(C) \da [\spec k / \Aut(C)]$.

:::

:::{.proposition title="Minimal presentations"}
Let $\cX$ be a Noetherian algebraic stack and $x\in \abs{\cX}$ a finite type point with smooth stabilizer.
Then there exists a diagram

\[\begin{tikzcd}
	{\spec k(u)} && {U\in\Sch} \\
	\\
	{\cG_x} && \cX
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHNwZWMgayh1KSJdLFsyLDAsIlVcXGluXFxTY2giXSxbMCwyLCJcXGNHX3giXSxbMiwyLCJcXGNYIl0sWzAsMV0sWzAsMl0sWzIsM10sWzEsM10sWzAsMywiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d)

Here $U\to \cX$ is smooth of relative dimension $\dim G_x$.
:::

:::{.corollary}
$\cX$ is DM iff stabilizers are finite and reduced.
:::

:::{.proof}
Set the top-left corner to an orbit $\OO(u)$, show $\OO(u) \to \cG_x$ is smooth, and use the flat slicing criterion.
:::

:::{.theorem title="Local structure"}
Let $\cX$ be a separated Noetherian DM stack and $x\in \abs{\cX}$ a finite type point with geometric stabilizer $G_x$.
Then there exists a nice etale presentation: $( [\spec A/G_x], w) \to (\cX, x)$ which is etale and affine, a quotient of an affine by a finite group, inducing an isomorphism of stabilizer groups at $w$.
:::

:::{.definition}
A morphism $\pi: \cX\to X$ from an algebraic stack to an algebraic space is a coarse moduli space if

1. $\pi$ is universal for maps to algebraic spaces, so $\cX\to Y \implies \exists! X\to Y$, and
2. $\forall k = \kbar$, $\cX(k)\modiso \to X(k)$ is bijective.

:::

:::{.remark}
Idea: remove stabilizers from stack in exchange for giving up a universal family.
Under mild conditions, (2) implies a bijection on topological spaces.
Condition (1) is analogous to being a categorical quotient, while (2) is analogous to $X$ being an orbit space.
:::

:::{.theorem}
If $G\actson \spec A$ is a finite group action on an affine scheme, then $[\spec A/G] \to \spec A^G$ is a coarse moduli space.
:::

:::{.theorem title="Keel-Mori"}
Let $\cX$ be a separated and finite type DM stack over a Noetherian ring.
Then $\exists \pi: \cX\to X$ a coarse moduli space such that 

1. $\pi$ is a proper universal homeomorphism,
2. $\OO_X \cong \pi_* \OO_{\cX}$,
3. stable under flat base change.

:::

:::{.remark}
An application: consider $\Mgbar$.
Because $\Aut(C)$ is finite and reduced, this is a DM stack.
Semistable reduction implies it is proper.
The Keel-Mori theorem implies existence of a coarse moduli space which is a proper algebraic space.
Proving projectivity is substantially more difficult.
:::

## Quasicoherent sheaves

:::{.remark}
In particular for algebraic stacks, one can form $\QCoh(\cX)$ and the standard adjunctions $f^*$ and $f_*$ where the latter is proper pushforward.
If $G\actson \spec A$ is an algebraic group action, then there is a diagram

\[\begin{tikzcd}
	{\spec A} && {[\spec A/G]} && {\B G} \\
	\\
	&& {\spec A^G}
	\arrow["p", from=1-1, to=1-3]
	\arrow["q", from=1-3, to=1-5]
	\arrow["\pi", from=1-3, to=3-3]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHNwZWMgQSJdLFsyLDAsIltcXHNwZWMgQS9HXSJdLFs0LDAsIlxcQiBHIl0sWzIsMiwiXFxzcGVjIEFeRyJdLFswLDEsInAiXSxbMSwzLCJcXHBpIl0sWzEsMiwicSJdXQ==)

Then 

- $q_* M = M$ forgets the $A\dash$module structure,
- $p^* M = M$ forgets the $G\dash$action,
- $\pi_* M = M^G$ recovers the invariants.

:::

:::{.definition}
 A good moduli space is a map $\pi: \cX\to X$ from an algebraic stack to an algebraic space such that 

1. $\pi_*$ is exact on quasicoherent sheaves, and
2. $\pi_* \OO_{\cX} \cong \OO_X$.

:::

:::{.remark}
If $\cX = [U/G]$ with $G$ finite reductive, this is equivalent to $U\to X$ being a good quotient.
:::

:::{.example}
If $G\actson \spec A$ is linearly reductive, then $[\spec A/G]\to \spec A^G$ is a good moduli space.
:::

:::{.theorem}
If $\cX$ is an algebraic space with mild hypotheses, then $\pi: \cX\to X$ is a good moduli space iff etale locally it looks like taking invariants, i.e. there are diagrams

\[\begin{tikzcd}
	{[\spec A/G]} && \cX \\
	\\
	{\spec A^G} && X
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJbXFxzcGVjIEEvR10iXSxbMiwwLCJcXGNYIl0sWzIsMiwiWCJdLFswLDIsIlxcc3BlYyBBXkciXSxbMCwxXSxbMywyXSxbMSwyXSxbMCwzXSxbMCwyLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=)

:::

:::{.theorem}
Let $\pi: \cX\to X$ be a good moduli space. 
Then

- $\pi$ is surjective.
- If $Z_i$ are closed and disjoint substacks of $\cX$, their images under $\pi$ are closed and disjoint.
- Closed points of $X$ correspond to closed points of $\cX$.
    Two orbits are identified iff their closures intersect.
- If $\cX$ is finite type over a Noetherian scheme then $X$ is finite type.
- $\pi$ is universal for maps to algebraic spaces.

:::