Questions tagged [stacks]
In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.
513 questions
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Properness of quotient map
I am new to algebraic spaces and stacks. My question is the following:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Let $[X/G]$ be the quotient stack. Then when the natural map $\pi: ...
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82
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finiteness of quotient map
I am new to algebraic space s and stacks. My question is the following:
Let $X$ be a scheme and $G$ be a group scheme acting on $X$. We have the natural map $\pi : X \to X/G$. When $\pi$ will be a ...
- 613
6
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1
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effective descent of coherent sheaves
I am new to stacks and algebraic spaces. I have the following question:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...
- 613
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145
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Bundles on stacks
We want to define what is a morphism of bundles over an algebraic stack. If $X$ is an algebraic stack and $V_n$ is the stack of rank $n$ vector bundles, a vector bundle on $X$ will be a morphism of ...
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Two elliptic curves with the same j-invariants
This is an interesting observation of mine when exploring moduli of elliptic curves.
Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
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4
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Grothendieck construction on fibred categories/stacks
This question is related to a previous question of mine, which has so far gone unanswered.
For a fixed site $\mathcal C$, the fibred categories over $\mathcal C$ form a (strict) $2$-category, see here....
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Check that a Sheaf is Invertible Etale Locally
A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
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107
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Pull back in quotient stacks
I am started learning about stacks. I am reading Alper's notes Stacks and Moduli. I am getting difficulty to verify the following is a cartesian square. Can someone give some hint how to start? For ...
- 613
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152
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Descent of classifying stack
Let $X$ be a variety over $k$ and $G$ be a finite abelian group. Then we know that $H_{fppf}^{2}(X,G)$ is in bijective correspondence with isomorphism classes of $G$-banded gerbes.
Now we consider a ...
- 253
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103
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Representability of stack of finite maps between curves
I am interested in the following moduli problem: The moduli functor $\mathcal{F}$ has $T$-points:
a nodal $n$-pointed curve $C/T$ of genus $g$.
a nodal $b$-pointed curve $D/T$ of genus $h$.
a finite ...
- 223
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245
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Does automorphism of classifying stack come from automorphism of group?
Let $k$ be a field and $G$ be a group. For simplicity, let us assume $\operatorname{char}k=0$ and $G$ is a finite abelian group. We do not assume $k$ is algebraically closed. If there is an ...
- 253
1
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60
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Distribution of the marked points on the components of a stable n-pointed curve of genus zero
Let $\overline{M}_{0,n}$ be the fine moduli space of stable n-pointed curves of genus $g=0$. Let $[(D_{0},p_1,...,p_n)] \in \overline{M}_{0,n}$. Suppose that each component of $D_0$ contains at least ...
- 560
3
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134
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The étale fundamental group of an DM stack acting on a locally constant sheaf
Let $W$ be a connected and separated Deligne-Mumford stack of finite type over an algebraically closed field $k$, and $w\in W$ a geometric point. Consider the étale fundamental group $\pi_1^{ét}(W,w)...
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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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139
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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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61
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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 ...
- 105
3
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205
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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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348
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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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118
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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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56
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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 ...
- 2,907
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169
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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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144
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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 (...
- 5,701
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149
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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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254
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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}...
- 5,701
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98
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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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181
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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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180
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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 ...
- 426
11
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1
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403
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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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0
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155
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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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138
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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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1
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373
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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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0
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377
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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 $G$-...
user501319
11
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1
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Gerbes over finite fields
Let $k$ be a field with algebraic closure $\bar{k}$.
Recall that a gerbe over $k$ is an algebraic stack $\mathcal{G}$ over $k$ such that the groupoid $\mathcal{G}(\bar{k})$ is connected. We say that $\...
- 21.1k
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1
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226
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Morphisms of fibered categories which are compatible with the chosen cleavages
Let $\pi \colon F \rightarrow C$ and $\pi' \colon F' \rightarrow C$ be two fibered categories over the category $C$. A morphism from $\pi \colon F \rightarrow C$ to $\pi' \colon F' \rightarrow C$ is a ...
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1
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254
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Examples when algebraic 1-stack = derived enhancement?
Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?
Let me take an example from notes of Bertrand Toen, page 41 of https:/...
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6
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1
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402
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Anafunctors vs the plus construction
Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations
$$G(M) := \text{...
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127
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Covering a stack by an open substack that contains all points of finite type
Let $X = \text{Spec}A$ be an affine scheme. It is well known that if $U$ is an open subset which contains $\text{SpecMax}A$, then $U\supseteq X$. The previous statement generalizes to arbitrary ...
- 2,907
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76
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Function vanishing on the image of a morphism of algebraic stacks
Let $f: X\longrightarrow Y$ be a morphism of algebraic stacks, and assume I know the Krull dimension of the ring of global functions on $Y$ to be $n$. Assume that the stack $X$ has "stacky" ...
- 2,907
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237
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Non-invertible map between principal bundles only locally trivial in a supercanonical Grothendieck topology?
It is a truth universally acknowledged that a map between principal bundles must be in want of an inverse.
However, the construction of said inverse in the context of a more general site $(S,J)$ ...
- 35.4k
10
votes
1
answer
404
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2-completeness of stacks
I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...
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1
answer
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Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...
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6
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1
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633
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Proving Zariski descent
I want to understand why the functor $\mathscr{D}$
sending an affine scheme to its associated derived
$\infty$-category satisfies Zariski descent. My understanding
is that one has to show that given a ...
- 507
2
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0
answers
198
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Picard group of a DM-stack
I wonder if the followings are true. Let $k$ be a field, $G/k$ is a finite constant group scheme acting on an affine $k$-scheme $\mathrm{Spec}(A)$. Then a line bundle on the quotient DM stack $[\...
- 455
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156
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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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99
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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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238
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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 (...
- 21.1k
3
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283
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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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131
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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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