stack

Tags: #AG #todo #projects/review - Refs: - http://www.ams.org/notices/200304/what-is.pdf - https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf - Homotopy theory for stacks - Jarod Alpers: https://sites.math.washington.edu/~jarod/moduli.pdf - Links: - Classical ideas: - scheme - Moduli stacks - moduli stack of elliptic curves - moduli stack of Higgs bundles - Unsorted/quotient stack - Gerbe - Algebraic space - [Artin stack\] - Deligne-Mumford stack - See category fibered in groupoids - proper morphism - stackification - Homotopy quotient - How to realize a stack as a homotopy quotient? - higher stack - K-theory - hypercovering - stable infinity category - foliated manifold - derived stack - gerbe

Stacks

Definitions

In terms of a pseudofunctor

attachments/Pasted%20image%2020220319213425.png

  • An algebraic space is a pair \((X, R)\) with \(R \subseteq X{ {}^{ \scriptscriptstyle\times^{2} } }\) an equivalence relation whose projections \(p_i: R\to X\) are etale morphisms. Idea: replace being locally isomorphic to affine space in the Zariski topology with the finer etale topology.

  • A prestack is a functor \({\mathsf{Aff}}{\mathsf{Sch}}_{/k}^{\operatorname{op}}\to {\mathsf{hoType}}\)

  • The prestack of quasicoherent sheaves over ${\mathsf{Sch}}_{/ {S}} $ is a stack wrt the fpqc topology.

  • A stack is a functor $M: \mathsf{C}\to {\mathsf{Sch}}_{/ {S}} $ that satisfies effective descent.

  • A 1-stack of groupoids on \(\mathsf{C}\) is a category fibered in groupoids \({\mathcal{X}}\to \mathsf{C}\) satisfying certain descent conditions. This form a category ${\mathsf{St}}^1 \leq { \underset{\infty}{ \mathsf{Cat}} }{} _{/ { \mathsf{C} }} $, so morphisms are cones (of functors) over \(\mathsf{C}\). For morphisms, \(f\simeq g\) means there is a natural transformation from \(f\) to \(g\) commuting with the projections to \(\mathsf{C}\), so one can form a homotopy category f 1-stacks.

  • A smooth proper stack is essentially a compact orbifold.

Artin Stack

attachments/Pasted%20image%2020220228092934.png

Algebraic Stacks

attachments/Pasted%20image%2020220323191326.png

attachments/Pasted%20image%2020220220034958.png

Sheaves on Stacks

attachments/Pasted%20image%2020220228093141.png

Results

  • Working with a moduli stack instead of a moduli space allows pretending the space is smooth and admits a universal family.

  • Neat trick from algebraic geometry: for a stack \(X/G\) where $X \in{\mathsf{Var}}_{/ {{\mathbb{C}}}} $ and \(G \in {\mathsf{Fin}}{\mathsf{Grp}}\), \begin{align*} H^*(X/G; {\mathbb{Q}}) \cong H^*(X; {\mathbb{Q}})^G \end{align*} where the RHS denotes the taking the \(G{\hbox{-}}\)invariants. Seems to only work over \({\mathbb{Q}}\). The quotient is scheme-theoretic. The actual definition involves equivariant cohomology.

  • $\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{M}\mkern-1.5mu}\mkern 1.5mu_{0, 4}\cong {\mathbb{P}}^1_{/ {{\mathbb{C}}}} $ using the classical cross-ratio.

Lifting property

As in the case of the cotangent complex:

attachments/Pasted%20image%2020220319213617.png attachments/Pasted%20image%2020220319213700.png

Examples

Quotient stacks and Picard stacks, in terms of torsors: attachments/Pasted%20image%2020220319213511.png

Exercises

  • Show that \({\mathbb{A}}^1_{/ {k}} /{\mathbb{Z}}\) for \(\operatorname{ch}k = 0\) is an algebraic space that is not quasiseparated.
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