--- created: 2023-04-03T11:57 updated: 2024-04-19T16:18 aliases: - stack - stacks - stacks MOC --- --- - Tags: - #AG #MOC #higher-algebra/DAG #AG/moduli-spaces - Refs: - [http://www.ams.org/notices/200304/what-is.pdf](http://www.ams.org/notices/200304/what-is.pdf) - [https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf](https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf) - [Homotopy theory for stacks](https://arxiv.org/abs/math/0110247) - Jarod Alpers: - - - Course on algebraic stacks: and #resources/full-courses - [Seminar notes on stacks](https://etale.site/livetex/stax.pdf) - [Masterclass: deformation theory, algebraic stacks](https://qgm.au.dk/video/mc/triple-masterclass/index.html) - Noncommutative stacks, DAVID AYALA, AARON MAZEL-GEE, AND NICK ROZENBLYUM: - Links: - Classical ideas: - [scheme](Unsorted/scheme.md) - [proper morphism](Unsorted/proper%20morphism.md) - [hypercovering](hypercovering.md) - [flat family](flat%20family) - [a stack is a category fibered in groupoids](Unsorted/a%20stack%20is%20a%20category%20fibered%20in%20groupoids.md) - [Quot schemes](Quot%20schemes) - [orbifold](Unsorted/orbifold.md) - [foliated manifold](foliated%20manifold) - Types of moduli spaces: - [fine moduli space](fine%20moduli%20space.md) - [coarse moduli space](coarse%20moduli%20space.md) - Types of moduli stacks: - [algebraic space](Unsorted/algebraic%20space.md) - [Deligne-Mumford stack](Deligne-Mumford%20stack) - [Artin stack](Artin%20stack.md) - [gerbe](Unsorted/gerbe.md) - Specific stacks: - [Unsorted/quotient stack](Unsorted/quotient%20stack) - [moduli stack of Higgs bundles](moduli%20stack%20of%20Higgs%20bundles) - [moduli space of curves](moduli%20space%20of%20curves) - [moduli stack of elliptic curves](moduli%20stack%20of%20elliptic%20curves.md) - [moduli stack of abelian varieties](moduli%20stack%20of%20abelian%20varieties) - [Hilbert scheme](Hilbert%20scheme.md) - [representation stack](representation%20stack.md) - Common uses: - [Gromov-Witten invariants](Unsorted/Gromov-Witten%20invariants.md) - [Lagrangian Floer 1ology](Unsorted/Lagrangian%20Floer%20homology.md) - [symplectic field theory](symplectic%20field%20theory) - [contact homology](contact%20homology) - [Fukaya categories](Unsorted/Fukaya%20category.md) - [string topology](Unsorted/string%20topology.md) - [stackification](stackification.md) - How to realize a stack as a [homotopy quotient](homotopy%20quotient)? - [level structure](level%20structure) - Projective moduli space - Separated moduli problem - [Flat family](Flat%20family.md) - [hilbert polynomial](hilbert%20polynomial.md) - [Unsorted/topological stack](Unsorted/topological%20stack.md) - In derived geometry: - [derived stack](Unsorted/derived%20algebraic%20geometry.md) - [stable infinity category](Unsorted/stable%20infinity%20category.md) - Tangent space to a functor - [cotangent complex](Unsorted/cotangent%20complex.md) - [loop stack](loop%20stack.md) - [compactly supported cohomology for quotient stacks](compactly%20supported%20cohomology%20for%20quotient%20stacks.md) - [moduli space](moduli%20space.md) - [motivation for stacks](motivation%20for%20stacks.md) - [[QCoh(B) is equivalent to Rep(G)]] - [[automorphisms necessitate stacks]] # stacks MOC Idea: stacks are geometrically modeled on [sites](Unsorted/site.md) $\cat S$, and e.g. $\Grpd$ is a stack modeled on $\cat S = \Set$ with the discrete topology. ![](attachments/Pasted%20image%2020220420101720.png) ![](attachments/Pasted%20image%2020220802130931.png) ![](attachments/Pasted%20image%2020220802131002.png) Write $D$ for the dual numbers. ![](attachments/Pasted%20image%2020220802131151.png) ![](attachments/Pasted%20image%2020220802131205.png) ![](attachments/Pasted%20image%2020220802131319.png) ![](attachments/Pasted%20image%2020220802131346.png) ## In moduli problems ![](attachments/Pasted%20image%2020220426013106.png) ![](attachments/Pasted%20image%2020220426013211.png) # Definitions ![](attachments/Pasted%20image%2020220508201920.png) Informal definition: ![](attachments/Pasted%20image%2020220421231305.png) In terms of a [pseudofunctor](pseudofunctor): ![](attachments/Pasted%20image%2020220319213425.png) - An **algebraic space** is a pair $(X, R)$ with $R \subseteq X\cartpower{2}$ 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 $\Aff\Sch_{/k}\op \to \hoType$ - Source: should interpret as the [infinity category](Unsorted/infinity%20categories.md) of derived rings over $k$...? - Target: the infinity category of spaces, i.e. [homotopy types](Unsorted/homotopy%20type.md). - The prestack of [quasicoherent sheaves](Unsorted/quasicoherent%20sheaf.md) over $\Sch\slice S$ is a stack wrt the [fpqc](fpqc) topology. - A **stack** is a functor $M: \cat C\to \Sch\slice S$ that satisfies [effective descent](Unsorted/effective%20descent.md). - A **1-stack of groupoids** on $\cat C$ is a [category fibered in groupoids](category%20fibered%20in%20groupoids) $\mcx \to \cat C$ satisfying certain [descent](Unsorted/descent.md) conditions. This form a category $\St^1 \leq \inftycat{} \slice{ \cat{C} }$, so morphisms are cones (of functors) over $\cat{C}$. For morphisms, $f\homotopic g$ means there is a natural transformation from $f$ to $g$ commuting with the projections to $\cat{C}$, so one can form a homotopy category f 1-stacks. - A smooth proper stack is essentially a compact [orbifold](orbifold.md). ## In terms of sheaves See [site](Unsorted/site.md). ![](attachments/Pasted%20image%2020220503235945.png) ![](attachments/Pasted%20image%2020220504000008.png) # Artin Stack ![](attachments/Pasted%20image%2020220228092934.png) # Algebraic Stacks ![](attachments/Pasted%20image%2020220323191326.png) ![](attachments/Pasted%20image%2020220220034958.png) # Geometric stacks As in the case of the [cotangent complex](Unsorted/cotangent%20complex.md): ![](attachments/Pasted%20image%2020220319213617.png) ![](attachments/Pasted%20image%2020220319213700.png) # Sheaves on Stacks ![](attachments/Pasted%20image%2020220228093141.png) # Quotient stacks ![](attachments/Pasted%20image%2020220508201947.png) # Examples - $\BG$: defined as $\torsors{G} \leq G\dash\Set$ in terms of [torsors](Unsorted/torsor.md). See [BG](Unsorted/classifying%20space.md), constructed as the [quotient stack](Unsorted/quotient%20stack.md) $[\pt/G]$. - For any $X\in G\dash\Set$, $\B GX = [X/G]$ whose objects are $\torsors{G}\slice{X} \leq G\dash\Set\slice{X}$. - $\Rep(G)$ can be interepreted as a category of sheaves on the stack $\BG$. Quotient stacks and Picard stacks, in terms of [torsors](Unsorted/torsor.md): ![](attachments/Pasted%20image%2020220319213511.png) # Exercises - Show that $[\AA^1\slice k/\ZZ]$ for $\characteristic k = 0$ is an algebraic space that is not [quasiseparated](Unsorted/separated.md). # Results - Working with a moduli stack instead of a [moduli space](moduli%20space.md) allows pretending the space is smooth and admits a universal family. - Neat trick from algebraic geometry: for a stack $X/G$ where $X \in\Var\slice \CC$ and $G \in \Fin\Grp$, $$ H^*(X/G; \QQ) \cong H^*(X; \QQ)^G $$ where the RHS denotes the taking the $G\dash$invariants. Seems to only work over $\QQ$. The quotient is scheme-theoretic. The actual definition involves [equivariant cohomology](equivariant%20cohomology.md). - fpqc stack $\implies$ fppf, etale stack