# Questions tagged [stacks]

In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.

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### Extending Tannakian "dictionary" to gerbes

The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories". Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
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### Do automorphisms actually prevent the formation of fine moduli spaces?

I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
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### Is the classifying stack of an abelian variety separated?

If $G/k$ is an algebraic group, a necessary condition for $BG$ to be separated over $k$ is that $G$ is proper over $k$. Assuming that $G$ is smooth and connected we are reduced to the case of an ...
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### Why this morphism of stacks is an isomorphism?

Let $C$ be a site and $p : F \mapsto C$ $p' : F' \mapsto C$ two categories fibred in groupoids over $C$ which are stacks (see, e.g., the definition here https://stacks.math.columbia.edu/tag/0268). For ...
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### Stack of smooth fiber bundles with fiber $F$

I'd like to premise that while I know the definition of (differentiable) stack, I'm not really into the language of schemes so my understanding of what is a moduli stack is pretty concrete and ...
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### Category of sheaves of vector spaces on BG

Let $G$ be an affine group scheme over $\mathbb{C}$. I am interested in understanding the differences between different notions of sheaves on the stack $pt/G = BG$. For any algebraic stack $X$ one can ...
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### What, precisely, is a stratification of a stack?

I'm currently working on a structure result for a certain spectral moduli problem, and I've been running into the problem of having to define what, precisely, is meant by the term "stratification&...
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### Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid

This questions is about the distinction between: Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...
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### Frobenius pullback of an integrable connection on a quasi-projective scheme

Let $X_k$ be a smooth quasi-projective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to characteristic $0$, and let $(X_K)^{\text{an}}$ denote the rigid analytic space associated ...
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### Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks

I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...
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### Nice proof that de Rham complex computes Lie algebra cohomology?

If $G$ is a nice enough group acting on a nice enough space $X$, then the relative de Rham complex $$\Omega^\bullet_{X/(X/G)}\ \simeq\ \mathcal{O}_X\otimes\text{Sym}\,\mathfrak{g}^*[-1]$$ is given by (...
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### Relationship between $\infty$-categories of ind-coherent sheaves on the base and total space

My question is vaguely as follows, let $G\to E\to X$ be a principal $\infty$-bundle for some group object $G$ in a category of ($\infty$-)stacks. Can we recover the category $\operatorname{IndCoh}(E)$ ...
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### Dualizing sheaf for classifying stack and duality

For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...
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### Moduli stack of l-adic sheaves?

Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors. Let $\ell$ be a prime ...
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### Grothendieck purity for Brauer groups of stacks

Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
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### Infinitesimal criteria for unramified morphism on stacks

In an Artin stack, the tangent complex is fundamentally a derived object, often viewed as a 2-term complex via some quotient presentation, as discussed in Raskin's note here, for example. In ...
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### What is bad when stabilizers are non-reductive in moduli stacks?

Here is J. Alper's definition of good moduli spaces. Consider in characteristic zero. Then we see that the classifying stack of any non-reductive group $H$ does not have a good moduli space. In ...
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### How to differentiate natural transformations?

Let $G_{\bullet} = (G_1 \rightrightarrows G_0)$ and $H_{\bullet} = (H_1 \rightrightarrows H_0)$ be Lie groupoids and $\varphi_\bullet, \psi_\bullet : G_\bullet \to H_\bullet$ be Lie groupoid morphisms,...
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### G-local systems via the classifying stack BG

First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...
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### Moduli of morphisms between varieties

Let $\mathcal{M}, \mathcal{N}$ be two "well-behaved" (i.e. representable by an algebraic stack parametrising flat, proper, surjective, finitely presented morphisms with geometric fibres ...
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### Stacks v.s. Stratifolds?

Both stacks and stratifolds can model spaces with singularities (say, singular quotient space for example). In my little experience, stacks are widely used in moduli problems and stratifolds are often ...
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### Local structure of smooth morphisms of stacks

Let $\varphi:X \to Y$ be a smooth morphism of schemes. There is a well-known local structure theorem: Zariski locally $\varphi$ is given by the composition of an etale morphism and an affine space (...
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### When a stack quotient coincides with GIT quotient?

Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive. Question: Is it true that when $G/H$ is open in its affine ...
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### Computing the equivariant Chern character

Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ ...
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### Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves

I am reading this paper https://arxiv.org/abs/1608.04797 Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh_P$ be the ...
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Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\... • 2,103 2 votes 0 answers 120 views ### Moduli stack of doubly periodic complexes? Let$\mathcal{A}$be an abelian category. In HAG II Toen and Vessozi built a higher derived stack$X$whose category of perfect complexes is$\text{Perf}(X)\simeq D^b(\mathcal{A})$. So$X$is a good ... • 5,565 7 votes 1 answer 353 views ### Faithfully flat descent in complex analytic geometry A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "... • 543 5 votes 1 answer 339 views ### How the automorphism group of an elliptic curve acts at the localization of the stack$\mathcal{M}_{1, 1, k}$at the corresponding point I am studying the enlightening article "The Picard Group of$\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof. Setting Let$\mathcal{M}_{1, 1, k}$... • 53 2 votes 0 answers 394 views ### Understanding the universal family$\mathcal{M}_{g,1}\to\mathcal{M}_g$I am trying to understand why the morphism of prestacks$\mathcal{M}_{g,1}\to\mathcal{M}_g$is the "universal family". Here$\mathcal{M}_g$is the moduli stack of curves of genus$g$, its ... • 1,061 6 votes 1 answer 463 views ### Completion of the classifying stack$BG$at a point With the classifying stack$BG$I have come across "the formal completion of$BG$at point", which is denoted$\widehat{BG}$, for instance on page 7 of https://arxiv.org/pdf/1703.08578.pdf, ... 3 votes 0 answers 162 views ### Linear deformations of a morphism between stacks Given smooth algebraic stacks$\mathcal{X}$,$\mathcal{Y}$what are the linear deformations$\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$of a morphism$f:\mathcal{X} \to \mathcal{Y}$? In ... 1 vote 0 answers 311 views ### Deformation theory of stacks and the tangent complex On a smooth stack X one can construct the two-term tangent complex$T_X \in D(X)$, as in Sam Raskin's notes (https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf). I ... 6 votes 1 answer 375 views ### Fibers of the coarse moduli space map Let$\mathcal{X}$be a Deligne-Mumford stack over a field$k$which admits a coarse scheme$c : \mathcal{X}\rightarrow X$. This will be the case if$\mathcal{X}$is separated and locally of finite ... • 7,313 1 vote 0 answers 128 views ### Extending Grothendieck topology from analytic manifolds to spaces? Let$k\text{-Man}$denote the Euclidean site of$k$-analytic manifolds where$k=\mathbb{R}, \mathbb{C}$. In words,$k\text{-Man}$is the usual category of real/complex analytic manifolds considered ... • 2,890 0 votes 0 answers 185 views ### What happens if you apply$\operatorname{QCoh}$to$(X,\underline{\mathbb{C}}_X)$?$\DeclareMathOperator\QCoh{QCoh} \newcommand\numC{\mathbb{C}} \newcommand\numQ{\mathbb{Q}}$Let$X$be a topological space (possibly a scheme). What happens if you apply$\QCoh$to the prestack$(X,\...
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The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...