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Questions tagged [stacks]

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2
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0answers
70 views

Central extension gives a gerbe over stack

Consider a central extension of Lie groups $1\rightarrow S^1\rightarrow \hat{G}\xrightarrow{\pi} G\rightarrow 1$. I understand that this mean $\pi:\hat{G}\rightarrow G$ is a surjective homomorphism ...
6
votes
0answers
54 views

Does the category of $G$-equivariant sheaves have enough injectives?

The question is related to this one. Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$. Let $G$ be a topological group which ...
2
votes
1answer
180 views

Understanding the definition of $G$-gerbe

In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following. Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\...
0
votes
1answer
84 views

Fibered product of stacks comes from a Lie groupoid

Suppose $\mathcal{G},\mathcal{H}$ are Lie groupoids and $F:B\mathcal{G}\rightarrow B\mathcal{H}$ be a morphism of stacks. We can talk about the fibered product $B\mathcal{G}\times_{B\mathcal{H}}B\...
3
votes
0answers
121 views

Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable

I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu. Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open ...
3
votes
1answer
172 views

Extend Group Action of $\mathbb{A}^1 /G$ to Projective Line

My question refers to an argument used in Torsten Ekedahl's paper: https://arxiv.org/abs/0903.3148 in Example ii) (page 8): We consider a finite subgroup of affine transformation of $\mathbb{A}^1$. ...
2
votes
0answers
92 views

Is there a definition of an unpointed schematic homotopy type?

In the paper Champs Affines (http://www.math.univ-toulouse.fr/~btoen/chaff.pdf) Toen introduces pointed schematic homotopy types (SHTs) to solve Grothendieck's schematization problem (described in ...
8
votes
4answers
604 views

Motivation for definition of Quotient stack

I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows. Let $G$ be a Lie group action on a manifold $X$ (left action). ...
14
votes
1answer
861 views

GAGA for stacks

I am curious about stacky generalizations of the following GAGA theorem: If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is ...
4
votes
2answers
171 views

unit element under map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$

Let $\mathcal{G}$ be a Lie groupoid. The target map $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a principal $\mathcal{G}$ bundle. This article Orbifolds as Stacks? by Eugene Lerman calls (in page $...
4
votes
1answer
230 views

Stack being represented by a scheme/manifold

On page $10$ of the survey article Algebraic stacks, by T. Gomez (arXiv:math/9911199), we have following result If a stack has an object with an automorphism other than the identity, then the ...
7
votes
2answers
227 views

$2$-fiber product is a scheme then map of stacks is representable

Ariyan Javanpeykar said here in comments that, $X\times_{\mathcal{X}}X$ being a scheme is equivalent to representability of $X\rightarrow \mathcal{X}$. Context is as in this question. Suppose $p:...
2
votes
0answers
143 views

Diagonal is representable then composition is representable

Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...
3
votes
1answer
277 views

Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that, If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable? I could not find the statement (...
2
votes
0answers
107 views

English translation of G.Laumon, L.Moret-Bailly book Champs algébriques

Is there an English translation of G.Laumon, L.Moret-Bailly book Champs algébriques. Most questions on this site on stacks received this book as reference in comments/answers. So, I want to ask if ...
5
votes
1answer
342 views

$BG$ the stack, $BG$ the simplicial presheaf

I have a theoretical question about comparing two objects that I have recently come across. For concreteness, let us work over the category $C$ of schemes over $k$. Let $G$ be an algebraic group over ...
3
votes
0answers
199 views

from functor of points to stacks (or almost)

I want to have a better "working knowledge" of stacks. However every time that I approach the topic, I finish hitting a wall of technical details. They are essential, but I don't need them for now. ...
5
votes
1answer
215 views

What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space?

Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space. ...
0
votes
0answers
37 views

Confusion in the definition of epimorphism of morphism of categories fibered in groupoids

I am reading Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu. According to this paper, a morphism of groupoid fibrations (Categories fibered in groupoids over the category of manifolds) $...
4
votes
0answers
111 views

fully faithful Fourier-Mukai for stacks

https://arxiv.org/abs/math/9809114 Theorem 1.1 gives a fiberwise criterion for a Fourier Mukai functor to be fully faithful. I am looking for a similar result on stacks with the maps being not ...
0
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0answers
59 views

Lie groupoid with non trivial automorphisms does not come from genuine space

This question came from comments of this question. Suppose $\mathcal{G}$ is a Lie groupoid, we consider the category $B\mathcal{G}$ whose objects are principal $\mathcal{G}$ bundles and morphisms are ...
7
votes
2answers
389 views

Understand the difference between two stacks

Let us work over $\mathbb{C}$. Let $G$ be a finite group, acting on $\mathbb{A}^1$ via a character, and let $H$ be the kernel of the action. Assume that $\mathbb{A}^1$ is the coarse moduli space of ...
2
votes
1answer
226 views

Is a gerbe over a manifold is a special case of a gerbe over a stack?

There is a notion of Gerbe over a Manifold and a notion of Gerbe over a stack. Given a manifold $M$, there is a way to associate a stack $\underline{M}$ with it and this gives an embedding of ...
2
votes
1answer
262 views

sheaf cohomology and deRham cohomology of a stack

I am reading https://arxiv.org/pdf/math/0605694.pdf to understand about (hyper)cohomology groups of stack $\mathcal{X}$ with valued in a complex of abelian sheaves $\mathcal{M}$. Let $\mathcal{F}$ be ...
1
vote
1answer
128 views

Cohomological description of gerbes over stacks (orbifolds)

When understanding about gerbe over a manifold $X$ from Hitchin - Lectures on special Lagrangian submanifolds it is said that We are basically in gerbe territory (for smooth manifolds) if any one ...
0
votes
0answers
105 views

Differential gerbes as groupoid extensions

I started reading Non abelian differential gerbes https://arxiv.org/abs/math/0511696v5 It says in abstract : We study non-abelian differentiable gerbes over stacks using the theory of Lie ...
3
votes
2answers
466 views

Understanding the definition of atlas of a stack

I am reading https://arxiv.org/abs/0806.4160 to understand orbifolds as stacks. Definition : Let $D\rightarrow Man$ be a stack over category of manifolds. An atlas for $D$ is a manifold $X$ and a ...
2
votes
1answer
194 views

When can you resolve rational maps to proper stacks by blowing up?

Let $X$ be a surface over a field $k$ with a smooth $k$-point $x\in X$, and suppose $\mathcal{Y}$ is a proper DM stack over $k$. (I am really thinking of $\mathcal{Y}=\overline{\mathcal{M}_g}$.) Let $...
1
vote
4answers
187 views

Composition of bibundles

I am reading Orbifolds as stacks? Given Lie groupoids $\mathcal{G}$ and $\mathcal{H}$ there is a notion of what is called a bibundle from $\mathcal{G}$ to $\mathcal{H}$ which is supposed to be a ...
8
votes
2answers
404 views

Moduli 'space' of stacks?

In algebraic geometry, we are frequently interested in parametrizing geometric objects. Formally, parametrization of geometric objects having some property can be viewed as a functor $F:Sch\rightarrow ...
10
votes
0answers
317 views

What is a derived Kähler manifold?

From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space. Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open ...
2
votes
0answers
188 views

Tangent and cotangent bundle of a smooth algebraic stack

Are there any good notion of tangent and cotangent bundle (and stacks) of a smooth algebraic stack, similar to the notion of tangent and cotangent bundles (and spaces) of smooth schemes? I am ...
3
votes
1answer
162 views

Stack descent to sheaf descent via Grothendieck construction?

Let S be a Grothendieck site, the (either left or right adjoint to the) Grothendieck construction assigns to each groupoid fibration over S a presheaf valued in groupoids. The following feels it might ...
3
votes
1answer
201 views

Reduction of structure group for stacks

Consider an action of a smooth linear algebraic group $G$ on a variety $X$ over an arbitrary field $k$, and the quotient stack $[X/G]$. Let $p$ be a $k$-point of $X$. If the action is transitive (i.e. ...
5
votes
0answers
152 views

closed substack of quotient stack

The question concerns quotient stacks. I am not very comfortable with stacks, so feel free to edit the question if I am saying nonsense. It is also the reason why I may be spelling in too much detail ...
9
votes
0answers
244 views

What is there in the book Cohomologie non abélienne by Jean Giraud

These days I am trying to understand about stacks and gerbes. Most of the articles that has something to do with gerbes cite this work Cohomologie non abélienne by Jean Giraud. I do not read the ...
3
votes
0answers
152 views

What properties does the inertia stack $I_X$ of an algebraic stack $X$ inherit from $X$?

Let $S$ be a noetherian scheme and let $X$ be a "nice" algebraic stack over $S$. For instance, let's say $X$ is a finitely presented algebraic stack over $S$, or that $X$ is a finite type separated DM-...
5
votes
0answers
161 views

“Strict” homotopy theory of topological stacks/orbifolds

If we fix a finite group $G$, there are two different useful homotopy theories on the set of $G$-equivariant topological spaces (which are CW complexes, say). One, the "weak" homotopy theory, is given ...
7
votes
2answers
573 views

Derived topological stacks?

I apologize for the vagueness of the following. Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory ...
2
votes
1answer
249 views

Common gerbes over two K3 surfaces

Let $X$ and $Y$ be K3 surfaces over the complex numbers. Under what assumptions, do there exist a finite group $G_X$ a finite group $G_Y$ a $G_X$-gerbe $\mathcal{X}\to X$ (for the fppf topology) a $...
6
votes
1answer
410 views

Commutative group algebraic spaces

Is the category of commutative group algebraic spaces (commutative group objects in algebraic spaces) locally of finite type over a field, an abelian category? I would benefit from a reference
5
votes
0answers
130 views

Is the stack of smooth canonically polarized surfaces uniformisable (or a good orbifold)

This is a question about the fundamental group of the moduli of canonically polarized surfaces. Let $\mathcal{M}$ be the (locally finite type separated Deligne-Mumford) algebraic stack of smooth ...
6
votes
1answer
414 views

What's a (infinity-) semi-stack?

A stack is an object that mixes the notions of (algebraic) space and group. The key insight of stack theory is that most things you would want to do with spaces you can do with stacks: namely, you ...
2
votes
0answers
115 views

Irreducible components of $\partial \bar{H}_{g,n}$

Let $\bar{H}_{g,n}$ be the moduli space (or moduli stack) of $n$-pointed stable hyperelliptic curves. I am interested in understanding its rational Picard group. In the case $n=0$, the rational ...
4
votes
0answers
332 views

The lisse-etale site and derived algebraic geometry

If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the ...
9
votes
0answers
242 views

Do isomorphisms spread out under suitable assumptions?

I'm still trying to wrap my head around the "pathological" algebraic space $\mathbb{A}^1/\mathbb{Z}$; see Questions about the algebraic space $\mathbb{A}^1/\mathbb{Z}$ and Why is this not an algebraic ...
5
votes
0answers
420 views

Questions about the algebraic space $\mathbb{A}^1/\mathbb{Z}$

Let $X = \mathbb{A}^1_{\mathbb{C}}/\mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{A}^1$ via translation. [To clarify, $X$ is an \'etale sheaf with a smooth presentation $\mathbb{A}^1_{\mathbb{C}}\to ...
3
votes
0answers
242 views

Stacks algebraic over a given stack

Given a base site $S$ of schemes, consider a stack $\mathcal{C}$ on $S$ not assumed to be algebraic. Then in principle, given another stack $\mathcal{E}$ on $S$, together with a map of stacks $\...
10
votes
1answer
387 views

What is the total space of a stack after all?

From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be ...
9
votes
1answer
371 views

Universal homeomorphism of stacks and etale sites

A morphism between schemes is a universal homeomorphism if it is integral, surjective, universally injective. For morphism between algebraic stacks, this notion also make sense. It is well know that ...