Questions tagged [stacks]
The stacks tag has no usage guidance.
433
questions
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
0answers
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
$$\mathcal{F}...
2
votes
1answer
243 views
Sheaf of elliptic curves up to isogeny
For a scheme $X$, denote by $\mathcal{Ell}_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor
$$
\mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{...
6
votes
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
votes
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
votes
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
votes
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 ...
6
votes
0answers
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 ...
1
vote
0answers
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....
1
vote
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 ...
2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
20
votes
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 ...
1
vote
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 $\mathbf{QCoh}(X)$ ...
5
votes
0answers
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
votes
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
votes
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
votes
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
vote
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
votes
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 $\mathcal{F}$...
1
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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 ...
