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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connections on Lie groupoids/differentiable stacks
Let $(\Gamma_1\rightrightarrows \Gamma_0)$ be a Lie groupoid.
There are many places which define the notion of connection on a Lie groupoid.
As far as I have seen, there is no mention of these ...
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Reference request: Derived structure on the moduli stack of Higgs bundles
I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that
It is often better to put
derived ...
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2
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Do disjoint unions of stacks commute with finite fibre products?
Choose a big $\mathit{fppf}$-site $(\mathbf{Sch})_{\mathit{fppf}}$ and let $S$ be a scheme in that site.
Let $\{\mathcal{X}_i\mid i\in I\}$ be a family of stacks in groupoids over $S$ and let $\...
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Integrality of Atkin-Lehner operator for $\Gamma_1(N)$
A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. ...
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basic question on quotient stacks
Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of ...
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Limit of quotient stacks
Let $k$ be a field (we can set it to be either perfect or algebraically closed if necessary), let $G$ be a (split) reductive group over $k$. Let $(X_i)$ be a filtered projective system of finite type $...
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Smoothness of the stack of Shtukas without modifications
It is a well-known fact, that the stack of shtukas $\mathop{\mathcal{S}\!\mathit{ht}_{\pm 1}^1}$ with two legs and elementary modifications (i.e. those of type $(1,0,\dotsc,0)$ or $(0,\dotsc,0,-1)$) ...
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Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?
In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. ...
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Compactly supported cohomology of a topological DM stack
Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...
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What is the local structure of a general Artin stack?
Let $X$ be an Artin stack over the complex numbers. What can one say about the local structure of $X$, i.e. what is the simplest class of stacks by which were can always find a cover of $X$ by open ...
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Stacks as local quotients or via atlases
If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like:
A DM (resp. Artin) stack is a stack $X$ satisfying [insert condition in the diagonal ...
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3
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Lie groupoids in practice
I am familiar with the notion of Lie groupoids.
But, only easy examples of Lie groupoids I am familiar with are the following:
Lie groupoids coming from manifolds; that are of the form $(M\...
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Reference for calculating the dimension of algebraic stacks
I am interested in the dimension of algebraic stacks.But, I’m in trouble because I can’t calculate it.Are there any good reference or tools for calculating it?
I use the “Champs algébriques” by ...
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Spaces of Bordered (Ordinary, Spin- or Super-) Riemann Surfaces
It is known from the work of Deligne and Mumford that the "space" of punctured/marked Riemann surfaces is a Deligne-Mumford stack. I have few questions regarding similar statement for the spaces of ...
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$C^*$-algebras appearance in study of Lie groupoids and differentiable stacks
I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...
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Connectedness for stacks
Let $X$ be a stack for the Zariski (or etale) site over an arbitrary field $k$. The functor $\pi_0(X) : Alg_k \to Sets$ of path-components of $X$ is defined as the composition
$$Alg_k \overset{X}{\...
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Reducing the stack condition (descent condition) over an fpqc site to the case of single coverings
This is the lemma 4.25 of Vistoli's note
Let $S$ be a scheme, $\mathscr{F} \to \mathscr{S}ch/S$ a fibred category.
Then $\mathscr{F}$ is a stack over the fpqc site on $S$ iff
(1) $\mathscr{F}$ ...
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Significance of some expected results when defining Grothendieck topology
Let $\mathcal{C}$ be a category. Fixing an object $U$ of $\mathcal{C}$, there are some obvious functors we can associate to it, for example:
$h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ given by $V\...
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Stack associated to Groupoid object in category $\text{Sch}/S$
Consider the category of manifolds $\text{Man}$.
A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the ...
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stack (in groupoids) over a site $\mathcal{C}$
Question : What is a stack (in groupoids) over a site $\mathcal{C}$ for you?
There are two a ways to think about it.
A stack over a site $\mathcal{C}$ is a category $\mathcal{D}$ with a functor $\...
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Reference requence: scheme of complete homomorphisms of rank $r$ via blowups
I'm reading these notes
where it states in section $3$: (transcribed because I can't post image)
Step 1. Introduce the stacks of degenerated and iterated shtukas
which extends that of shtukas.
This ...
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Prerequisites for understanding algebraic geometry of “algebraic gerbes”
I am trying to learn about algebraic geometry of gerbes.
I am familiar with set up of gerbes in the case of differential geometry. Though there is some similarity between differentiable gerbes and ...
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Smooth sub-orbifolds in the language of stacks
In most geometric categories, "monomorphism" is too general to describe useful notions of "embedding". This is the case e.g. for schemes, complex manifolds, and differentiable manifolds.
So "embedding"...
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Road map for moduli space/moduli problem/moduli stack
I am familiar with (most of the) contents of Angelo Vistoli's notes on Descent theory (Stacks). I am also comfortable with basics of Schemes, their Cohomology (Cech), from Hartshorne's Algebraic ...
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Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe
Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
Context:
Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
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Stacks in moduli spaces of sheaves research
I want to get some practice and build more appreciation for the use of stacks in the context of classical moduli spaces of sheaves. Here by classical I vaguely mean hands-on description of the ...
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Surjectivity of pushforward on Chow rings for stacks
Let $f:X\rightarrow Y$ be a proper morphism of smooth Deligne-Mumford stacks of finite type over $\mathbb{C}$ that is birational, but not flat. The coarse spaces of $X$ and $Y$ are both not smooth. Is ...
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Which definition of "proper" is better?
It is well known that topology and algebraic geometry assign different meanings to the word "proper".
Let us recall the relevant definitions from topology (and we work in the context of topological ...
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Etale sheaves on algebraic spaces vs. Etale sheaves on affines
Let's fix a field $k$. First, consider $Aff_k$ to be the category of affine finite type $k$ schemes. On this category, one can define the etale topology and thus consider the site $Aff_k^{et}$, then ...
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Visualization and new geometry in higher stacks
I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond ...
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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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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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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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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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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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0
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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 ...
2
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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 ...
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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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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 ...
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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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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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$2$-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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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 ...
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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
answer
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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 ...