---
date: 2022-12-11 00:34
modification date: Sunday 11th December 2022 00:34:54
title: "Stacks and Moduli Study Guide"
aliases: [Stacks and Moduli Study Guide]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# References

-   Jarod Alper's course: <https://sites.math.washington.edu/~jarod/math582C.html>
-   Martin Olsson's, *Algebraic spaces and stacks*.
-   [What is a stack?](http://www.ams.org/notices/200304/what-is.pdf), by Dan Edidin.
-   [Equivariant geometry and the cohomology of the moduli space of curves](http://front.math.ucdavis.edu/1006.2364), by Dan Edidin.
-   [Notes on the construction of the moduli space of curves](http://front.math.ucdavis.edu/9805.5101), by Dan Edidin.
-   [Stacks for Everybody](http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf), by Barbara Fantechi.
-   [Picard groups of moduli problems](http://www.mathcs.emory.edu/~brussel/Scans/mumfordpicard.pdf), by David Mumford, and Daniel Litt's exposition thereof (parts [one](http://math.stanford.edu/~dlitt/exposnotes/picardI.pdf) and [two](http://math.stanford.edu/~dlitt/exposnotes/picardII.pdf)).
-   [Notes on Grothendieck topologies, fibered categories and descent theory](http://front.math.ucdavis.edu/0412.5512), by Angelo Vistoli (lovely comprehensive reference).
-   [More notes](https://stacky.net/files/written/Stacks/Stacks.pdf)

## Definitions

-   What is the [blowup of a scheme](blowup%20of%20a%20scheme)? The [blowdown of a scheme](blowdown%20of%20a%20scheme)?
-   What is a [site](site.md)?
-   What is a [topos](topos.md)?
-   What is the **Hilbert scheme**?
-   Topologies:
    -   What is the etale topology?
    -   What is the fppf topology?
-   What is the etale site?
-   What is an [algebraic space](algebraic%20space.md)?
-   What is [Faithfully flat descent](Faithfully%20flat%20descent)?
-   What is a [torsor](torsor.md)?
-   What is the Amtisur complex?
-   What does it mean to have a [closed subcategory](closed%20subcategory)?
-   What does it mean for a subcategory to be [local on the base](local%20on%20the%20base.md)? [local on the domain](local%20on%20the%20domain)?
-   What does it mean to have a [stable subcategory](stable%20subcategory)?
-   What is an etale equivalence relation?
-   What is a [locally closed](locally%20closed.md) substack?
-   What is an [orbifold](orbifold.md)?
-   What is a [smooth stack](smooth%20stack)?

## Results

-   What is the local criterion for flatness?
-   What is the infinitesimal criterion for flatness?

## Problems

-   [ ] Representable functors:
    -   [ ] What is \${`\mathbb{A}`{=tex}}\^n\_{/ {k}} \$ as a functor? What is it represented by?
        -   For schemes with structure sheaves taking values in $R{\hbox{-}}$algebras, `<span class="math display">     \begin{align*}      {\mathbb{A}}^n_{/ {{-}}}  \coloneqq{{\Gamma}\qty{({-}); {\mathcal{O}}_{{-}}} }{ {}^{ \scriptscriptstyle\times^{n} }  }: {\mathsf{Sch}}^{\operatorname{op}}\to {\mathsf{Alg}}_{/ {R}}  \\      X &\mapsto {{\Gamma}\qty{X; {\mathcal{O}}_X} }\carptpower{n} \\      f &\mapsto f^*      ,\end{align*}     <span>`{=html} which is represented by $\operatorname{Spec}{\mathbb{Z}}[x_0, \cdots, x_n]$.
    -   [ ] What is \${`\mathbb{P}`{=tex}}\^n\_{/ {k}} \$ as a functor? What is it represented by?
    -   [ ] What is ${\mathbb{A}}^n_{/ {k}} \setminus\left\{{0}\right\}$ as a functor?
        -   $x\mapsto \mathbf{x} = {\left[ {x_1, \cdots, x_n} \right]}$ where not all of the $x_i$ are zero.
-   [ ] Show that if \$X `\subseteq {\mathbb{P}}`{=tex}\^n\_{/ {k}} \$ is a closed subscheme and ${\mathcal{F}}$ is a coherent sheaf of ${\mathcal{O}}_X{\hbox{-}}$modules, then $\Gamma(X, {\mathcal{F}})$ is a finite dimensional $k{\hbox{-}}$vector space.
-   [ ] Construct a ring $R$ such that $\operatorname{Spec}R$ is \${`\mathbb{A}`{=tex}}\^1\_{/ {k}} \$ punctured at $0, 1$.
    -   $R = k[x] \left[ { \scriptstyle { { \qty{ x(x-1)} }^{-1}} } \right]$
-   [ ] Show that the cohomology of a quasicoherent sheaf on an affine scheme vanishes.