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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MSRI Workshop videos and lecture notes
I am referring to MSRI workshop "Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory".
There are videos and lecture notes available which can be seen on ...
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1
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Stack associated to Lie group and manifold
Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows *)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles.
Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)...
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2
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Yoneda Embedding and pull back
Given a manifold $M$ we have a geometric stack associated to it namely $\underline{M}$ whose objects are smooth maps to $M$. For the sake of consistency I am writing $BM$ for $\underline{M}$.
Given a ...
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1
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Sheaves over a sheaf
Everything I write I mean in the in the sense of Lurie's HTT.
Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C}...
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1
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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 ...
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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 ...
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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}\...
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2
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Fibered product of stacks comes from a Lie groupoid
I am adding some context here. I am reading Introduction to Differentiable Stacks by Gregory Ginot.
In page no $7$, just before the remark $2.2$ he says the following.
One shall be careful that ...
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3
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0
answers
156
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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 ...
- 6,023
2
votes
1
answer
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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 $\mathbb{A}^1$. ...
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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 ...
- 1,635
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4
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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 ...
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16
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1
answer
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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 ...
- 7,952
4
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1
answer
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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 $...
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1
answer
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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 ...
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2
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$2$-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:...
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0
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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 ...
- 6,023
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1
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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 (...
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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 ...
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1
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$BG$ 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 ...
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342
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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.
...
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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.
...
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4
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173
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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 ...
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2
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458
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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 ...
- 161
4
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1
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373
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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 ...
- 6,023
2
votes
1
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838
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Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves
I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$.
...
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2
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Cohomological description of gerbes over stacks
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 ...
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2
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1k
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Understanding the definition of atlas of a stack over the category of manifolds
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 ...
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1
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424
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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 $...
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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 ...
- 6,023
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2
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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 ...
user74900
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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 ...
user74900
2
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0
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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 ...
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1
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341
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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 ...
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1
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256
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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. ...
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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 ...
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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
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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-...
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"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 ...
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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 ...
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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 $...
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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
user119470
7
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1
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749
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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 ...
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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 ...
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552
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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 ...
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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 ...
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542
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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 ...
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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 $\...
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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 ...
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891
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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 ...
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