Questions tagged [stacks]

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7
votes
0answers
451 views

Understanding the higher stack of perfect complexes

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff: We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero ...
3
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1answer
167 views

Universal bundles over algebraic stacks

$\DeclareMathOperator\el{ell}$In the topological case, given a group $G$, we can define the classifying space $BG$ for principal $G$-bundles and there is a universal $G$-principal bundle $EG \to BG$ ...
3
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0answers
135 views

Do stacks with non-algebraic automorphism groups arise in nature?

Do stacks parametrizing objects with non-algebraic automorphism groups happen in nature e.g. in complex analysis or number theory? For example can the automorphism groups be non-projective complex ...
3
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0answers
165 views

Interesting stacks with affine space as coarse moduli

I am looking for examples of Deligne-Mumford stacks whose coarse moduli space is $\mathbb{A}^n$ or at least an open subscheme of $\mathbb{A}^n$ whose complement has codimension $2$. (Thus the whole ...
3
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1answer
166 views

Hom-space between Picard stacks

This is from Deligne, La formule de dualité globale, SGA 4, tome 3, Expose XVIII, and I am confused about how the hom-space between Picard stacks is again a Picard stack. A quick rewind. For a site $S$...
3
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0answers
145 views

Sheaf $\operatorname{Isom}(x,y)$ isomorphic to fibered product $U \times_{(x,y),X \times X, \Delta} X$

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Isom{Isom}$Let $S$ be a scheme and $C$ be the category $(Sch/S)$. Let moreover $p:X \to C$ be an algebraic stack over $C$. Consider an arbitrary ...
8
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1answer
277 views

Is ${\bf Set}$ the terminal autological topos

An autological topos is a type of topos defined by Mike Shulman in his paper on stack semantics; specifically, they are toposes satisfying an additional topos theoretic axiom schema expressed in their ...
4
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1answer
166 views

Do quotient stacks help classify the orbits of group actions on varieties?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting ...
5
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1answer
239 views

Construction of an atlas for the moduli stack $\mathcal{Bun}_X^{n,d}$ in F. Neumann's 'Algebraic Stacks and Moduli of Vector Bundles'

I'm reading Frank Neumann's "Algebraic Stacks and Moduli of Vector Bundles" and have some problems to understand a construction from the proof of: Theorem 2.67. (page 81) The moduli stack $...
5
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1answer
196 views

When quotient stacks (for nonsmooth group) are algebraic and related questions

Let $k$ be a field. Consider a group $k$-scheme $G$ and let $X$ be a $k$-scheme equipped with an action of $G$. Then one can define the quotient stack $[X/G]$. Objects of $[X/G]$ over $k$-scheme $T$ ...
2
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1answer
304 views

K/G-theory of affine bundles

Setting: $f : C \to D$ is a morphism of Artin stacks over $X$ which is a torsor for a vector bundle $T \to X$: étale-locally in $X$, we have $C \simeq D \times_X T$. I want to conclude that $f^*: G(D)...
5
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154 views

Étale stack on $\text{Spec}(k)$

Let $\mathcal{F}:\text{Ét}(\text{Spec}(k))^{\mathrm{op}}\rightarrow \text{Set}$ be an étale presheaf. The étale sheaf condition for the cover $\text{Spec}(l)\rightarrow \text{Spec}(k)$ is $$\mathcal{F}...
2
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1answer
243 views

Sheaf of elliptic curves up to isogeny

For a scheme $X$, denote by $\mathcal{Ell}_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor $$ \mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{...
6
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1answer
198 views

Formal completion of a quotient stack

$\newcommand{\Rep}{\operatorname{Rep}}$ $\newcommand{\mo}{\operatorname{-mod}}$ $\renewcommand{\hat}{\widehat}$ I apologize in advance if this is a naive question but my background in algebraic ...
7
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1answer
611 views

Stacks for a string theory student

First, I'm a string theory student hoping to grasp some math involved in some physics developments. I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the ...
8
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0answers
319 views

Geometric stacks, groupoids and étendues

If $(C, \tau)$ is a site with pullbacks and $\tau$ subcanonical, it is well known that these things are essentially equivalent: Groupoids $s,t: U_1 \to U_0$ where $s,t$ are covering for the $\tau$-...
9
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1answer
210 views

When is a twisted form coming from a torsor trivial?

Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P \times^G X := (P \times X)/\sim$ where $\sim$ is the ...
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211 views

Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space?

A version of Homotopy Hypothesis says that the Fundamental $n$-grupoids model Homotopy $n$-types... and if we continue upto $\infty$, then the Fundamental $\infty$- groupoids or Kan Complexes model ...
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177 views

Two definitions of cotangent complex

I have reading a paper by Professor Pridham(https://arxiv.org/abs/0905.4044v4). Page 47-48 contains a comparison of the two definitions of the cotangent complex, but there is a part I don't understand....
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1answer
202 views

Stabilizer $G_x$ of a $k$-valued point of an algebraic Stack

An algebraic stack or Artin stack is a stack in groupoids $\mathcal{X}$ over the étale site such that the diagonal map of $\mathcal{X}$ is representable and there exists a smooth surjection from (the ...
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0answers
86 views

Infinititesimal Automorphisms intuition (algebraic stacks)

Let $F$ be a category cofibered in groupoids over category $C$. Given a morphism $x'\to x$ in $F$ lying over a morphism $A′\to A$ in $C$, there is an induced homomorphism $\operatorname{Aut} A'(x')\to ...
5
votes
1answer
245 views

Fixed point stack for a torus action

In this paper, M. Romagny defines for an action of a group scheme $G$ on a stack $X$ the fixed point stacks $X^G$ associated to the group action on a stack and in Theorem 3.3 he proves that if the ...
5
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0answers
304 views

Definition of the cotangent complexes of Artin stacks

I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived ...
4
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0answers
132 views

The definition of flat sheaves of rings on algebraic stacks

Reading Sheaves on Artin stacks by M. Olsson I find this definition (3.7, (i)): Let $\mathcal X$ be an algebraic stack on a scheme $S$. A sheaf of rings $\mathcal A$ on $\mathcal X_{\textrm{lis-et}}...
2
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0answers
89 views

Representability result

Let $X$ and $S$ be schemes over a field $k$. Reading this paper, there is a result on the representability of a morphism (proposition 3.1, page 4). Which result or reference on representability is ...
2
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1answer
97 views

Is there any relation between two pseudofunctors associated to two different cleavages of the same fibered category?

It is well known that given a Fibered category $P_F: E \rightarrow C$ with a cleavage $K$ we can construct a pseudofunctor $F_K: C^{op} \rightarrow Cat$. Now if one chooses a different cleavage $L$ ...
3
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0answers
99 views

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 ...
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7answers
2k views

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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0answers
98 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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332 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 ...
3
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0answers
76 views

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 ...
8
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0answers
169 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 ...
2
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0answers
208 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 ...
1
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1answer
141 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, ...
15
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2answers
2k views

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 ...
4
votes
2answers
365 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 ...
5
votes
3answers
490 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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0answers
92 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 ...
3
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0answers
163 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 ...
3
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2answers
253 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
1answer
296 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 $M_k(\Gamma_0(N))$. ...
0
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1answer
169 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 ...
4
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0answers
212 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 $...
3
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0answers
71 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 $(0,\dotsc,0,-1)$) ...
4
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0answers
184 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 $\mathbb{C}$. ...
4
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0answers
134 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 ...
4
votes
1answer
490 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 ...
3
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1answer
364 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 ...
1
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3answers
306 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\...
2
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0answers
135 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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