# Questions tagged [stacks]

The stacks tag has no usage guidance.

407
questions

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### Coverings of (DM) stacks and categories of descent data

If $X$ is a DM stack, we know that there is a surjective étale morphism $U \to X$ with $U$ a scheme. Combining Lemma 4.5 of these notes and Proposition 12.7.4 of Champs Algébriques one should be able ...

**18**

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**7**answers

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### What are the occurrences of stacks outside algebraic geometry, differential geometry, and general topology?

What are the occurrences of the notion of a stack outside algebraic geometry, differential geometry, and general topology?
In most of the references, the introduction of the notion of a stack takes ...

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91 views

### Relationship between Quasicoherent sheaves and $\mathbb A^1$-fpqc modules over an fpqc stack

In what follows, assume several universes for simplicity.
Let $X$ be a stack in groupoids on the fpqc site of small affine schemes $\mathbf{Aff}_{\text{fpqc}}$. We can define $\mathbf{QCoh}(X)$ ...

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305 views

### What is the motive of $\operatorname{Bun}_G(X)$?

$\DeclareMathOperator\Bun{Bun}$Let $X$ be a scheme over algebraically closed field $k$, $G$ a reductive group and $\Bun_G(X)$ the stack of $G$ bundles on $X$. Write $[\Bun_G(X)]\in K_\text{st}$ for ...

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### How big is the complement of stable locus $\operatorname{Bun}G$

Let $\Sigma$ be a smooth projective curve, and $G$ a reductive group. Let $\operatorname{Bun}G$ be the stack of principal $G$ bundles on $\Sigma$ (with a fixed topological type).
What is the ...

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135 views

### What are the Newton groupoids from Drinfeld's paper on the Grinberg-Kazhdan theorem?

The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the ...

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157 views

### Why not consider categorical quotient in stacks?

Let's say we have a $k$ group scheme $G$ acting on a $k$ scheme $X$, we can consider its quotient in the category of stacks, the usual definition of it would be the quotient stack $[X/G]$ defined by
...

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**1**answer

122 views

### Are morphisms from affine schemes to Artin stacks affine morphisms?

It is explained in this MO discussion that if one has a morphism $f:X\rightarrow Y$ of schemes such that $X$ is an affine scheme, then $f$ need not be an affine morphism. However, if $Y$ is separated, ...

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### Understanding the definition of stacks

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...

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306 views

### Serre's theorem on global generations on stacks

Let $X$ be a quasi-projective scheme, the followings are quite useful.
Every coherent sheaf is globally generated after tensoring with a suitable line bundle.
Every coherent sheaf has trivial ...

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394 views

### How should one think about the band of a gerbe?

Let $X$ be a topological space. Let $\mathcal{F}$ be a fibered category over $X$; seen as an assignment of a category $\mathcal{F}(U)$ for each open $U\subseteq X$.
A fibered catgeory $\mathcal{F}$...

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80 views

### 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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141 views

### 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**answers

217 views

### 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 $\...

**3**

votes

**1**answer

276 views

### 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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126 views

### 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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176 views

### 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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66 views

### 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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172 views

### 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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121 views

### 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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421 views

### 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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332 views

### 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**answers

285 views

### 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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118 views

### 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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51 views

### 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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145 views

### $C^*$-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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145 views

### 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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357 views

### 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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201 views

### 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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174 views

### 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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76 views

### 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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**1**answer

92 views

### 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.
...

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338 views

### 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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137 views

### 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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294 views

### 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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222 views

### 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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205 views

### 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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124 views

### 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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**1**answer

215 views

### 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 $Aff_k^{et}$ and 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 manifolds ...

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253 views

### 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**answer

186 views

### 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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412 views

### 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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280 views

### 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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181 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 ...

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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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**1**answer

597 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}\...

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350 views

### 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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### 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 ...