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 (...
- 5,565
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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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Finite generation of stack cohomology
Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra?
For instance, $\text{H}^*(\text{B}\mathbf{G}...
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Tangent Space of Moduli of Log-Smooth Curves
We consider an algebraically closed field $\underline{k}$ and all constructions that we will consider are over this field. It is well known that for each relative nodal curve $\underline{f}: \...
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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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