All Questions
Tagged with stacks reference-request
41 questions
1
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56
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Frobenius pullback of an integrable connection on a quasi-projective scheme
Let $X_k$ be a smooth quasi-projective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to characteristic $0$, and let $(X_K)^{\text{an}}$ denote the rigid analytic space associated ...
- 2,907
2
votes
0
answers
98
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Tangent Space of Moduli of Log-Smooth Curves
We consider an algebraically closed field $\underline{k}$ and all constructions that we will consider are over this field. It is well known that for each relative nodal curve $\underline{f}: \...
- 223
11
votes
1
answer
403
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Grothendieck purity for Brauer groups of stacks
Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
- 388
8
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283
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Does Borel fixed-point theorem hold for Deligne-Mumford stacks?
Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus.
Question: Is the following statement true?
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7
votes
1
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507
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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 ...
- 8,524
9
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1
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1k
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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 ...
- 502
2
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0
answers
97
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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 ...
- 339
3
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0
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115
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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 ...
- 2,079
1
vote
1
answer
125
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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 ...
- 113
3
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0
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465
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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 ...
- 6,023
3
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0
answers
785
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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 ...
- 6,023
3
votes
0
answers
240
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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 ...
- 1,190
4
votes
0
answers
287
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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 ...
- 6,023
6
votes
1
answer
327
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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}...
- 595
4
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0
answers
1k
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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 ...
- 6,023
3
votes
0
answers
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.
...
- 943
4
votes
0
answers
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 ...
- 700
6
votes
1
answer
556
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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
2
votes
0
answers
132
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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 ...
- 21
3
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0
answers
289
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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 $\...
- 35.4k
3
votes
0
answers
146
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Localization of a 2-category
I am looking for a basic reference about localization of 2-categories, possibly avoiding the full formalism of n-categories.
- 2,384
2
votes
1
answer
481
views
Open and Dense Substack
I am looking for a definition of open and dense substack of a Deligne-Mumford stack $\mathcal X$. I have encountered this notion many times, but I am not able to find any references in which dense ...
- 123
3
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0
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277
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The symmetric product of a stack and its motive
This question is mainly a reference request.
I was wondering if there exists such a thing as the symmetric product of a quotient stack $[U/G]$, and in particular I would be interested in the motivic ...
- 430
1
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0
answers
136
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Is there a reference for boundedness of smooth canonically polarized varieties over Z (No...)
In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...
- 11
6
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0
answers
169
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How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?
It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...
- 9,009
4
votes
1
answer
289
views
Are automorphism groups of polarized varieties of finite type
It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal.
In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme $S$,...
- 171
1
vote
0
answers
191
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Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?
Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and
$$
X_n=\underbrace{ X\times_S \...
- 9,009
3
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0
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148
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Properties of the induced map between inertia stacks
Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to I\...
- 811
6
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2
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580
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Associating a principal bundle to a torsor
In Introduction to the language of stacks and gerbes, Moerdijk defines a torsor to be a sheaf $\mathcal{S}$ on $X$ with a freely transitive left-action of a sheaf of groups $\mathcal{G}$, such that $...
- 2,227
8
votes
1
answer
1k
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Quasi-coherent sheaves on classifying stacks
Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq \mathrm{...
- 63.1k
0
votes
1
answer
164
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Groupoid as a 2-coequaliser
Let $G=(G_1, G_0, s, t, u, i,\circ)$ be a groupoid, where $s, t$ are source and target maps, $i$ is the inverse, $u$ is the unit, and $\circ$ is the composition.
Denote $\underline{G_1}, \underline{...
- 1,271
2
votes
1
answer
258
views
picard group of moduli of elliptic r-prym curves
Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves $\overline{\mathcal{...
- 3,779
11
votes
2
answers
1k
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Diffeology as a sheaf on the site of smooth manifolds
Souriau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing ...
- 1,710
10
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2
answers
2k
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Reference for Weighted Projective Stacks
For a sequence of positive integers $a_0, \ldots, a_n$ and a ring $R$, there is a graded ring $R[x_0,\ldots, x_n]$ where $x_i$ is in degree $a_i$. There is a corresponding $\mathbb{G}_m$-action on $...
- 12.1k
6
votes
2
answers
1k
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Reference request: an algebraic stack whose closed points have no automorphisms is an algebraic space
The question is stated in the title. I think BCnrd states in a comment here
Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?
that while the answer is not found in Laumon & ...
- 2,218
1
vote
2
answers
946
views
Reference for moduli stack of principal G-bundles?
Hi,
I'm looking for a reference for the fact that the moduli stack $M_{GL_r,X}$ of $GL_r$-bundles over $X$ is an algebraic (Artin) stack. I'm only interested in the case where $X$ is a curve (for now)...
- 21k
10
votes
1
answer
760
views
Local structure of Deligne-Mumford stacks
Let $\mathcal{X}$ be a separated Deligne-Mumford stack over an algebraically closed field $k$ and let $X$ be the corresponding coarse moduli space, which we assume to exist. There is a map $p:\mathcal{...
- 101
3
votes
1
answer
729
views
References for constructible sheaves on complex analytic stacks
I'm looking for references on constructible sheaves and the six operation formalism on analytic stacks (stacks fibered over complex analytic spaces). Does anyone have some suggestions?
Basically I ...
- 4,265
8
votes
2
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584
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The Grothendieck plus construction for stacks of n-types
In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+2\right)$ times, and ...
- 15.5k
8
votes
1
answer
617
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Who first came up with the idea of essential/Morita equivalence of internal groupoids/categories?
The idea that stacks can be identified with groupoids internal to the base site $S$ up to what is variously called essential/Morita equivalence is well known. The basic idea is that one takes the 2-...
- 35.4k
65
votes
17
answers
17k
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Good introductory references on algebraic stacks?
Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...
- 1,538