# Questions tagged [algebraic-stacks]

The algebraic-stacks tag has no usage guidance.

169
questions

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### Operation on algebraic stack

Let $X$ be a smooth algebraic stack and $D$ be a normal crossing divisor. Let $\pi: \tilde{D}\rightarrow D$ be the normalization of $D$. Let $\sigma: \tilde{D}\rightarrow \tilde{D}$ be an involution. ...

**18**

votes

**6**answers

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

**1**answer

63 views

### Determine whether a rational function on the codomain of a surjective morphism is regular

Let $X$ be a smooth affine algebraic variety with a (not necessarily free) action by an algebraic torus $T$. Let $Y$ be the quotient stack $X/T$ and let $p:X\rightarrow Y$ be the quotient map. Suppose ...

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129 views

### Is there a stacky definition of irreducible symplectic manifold?

I am now interested in studying symplectic structures in the field of stacks.
In particular, is there a stacky definition of irreducible symplectic manifold ?
I'm also interested in similar things in ...

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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 $\mathbf{QCoh}(X)$ ...

**3**

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68 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 ...

**2**

votes

**1**answer

144 views

### Representaility of morphism of stacks for schemes

I have seen two definitions of representability of a morphism of stacks, which should be at least compatible with the definition of a morphism of categories fibered in groupoids.
(Representable ...

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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

121 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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145 views

### A quotient stack $[X/G]$ and irreducibility of $X$

Let $k$ be an algebraically closed field.
Let $X$ be a variety over $k$ and $G$ be a algebraic group over $k$ acting on $X$.
Question
If the quotient stack $[X/G]$ is irreducible, then is $X$ also ...

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**2**answers

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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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 ...

**7**

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**0**answers

261 views

### Stacky proof of no elliptic curves over Z

It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...

**4**

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162 views

### connection on principal bundles over algebraic/geometric stacks

Is there a notion of connection on a principal bundle over an algebraic or geometric stack?
By a geometric stack, I mean a stack over category of manifolds that is representable by a Lie groupoid; ...

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**1**answer

186 views

### Representable diagonal map $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for DM-Stacks/algebraic spaces

Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
has to be ...

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38 views

### Definition of Morphisms of algebraic stacks smooth of relative dimension n

There is a notion of smooth morphism of algebraic stacks e.g. Tag 075U and a notion of
relative dimension of a locally of finite type morphism $T\to \mathcal{X}$ from an algebraic space into an ...

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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 $(0,\dotsc,0,-1)$) ...

**4**

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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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**1**answer

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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117 views

### Perfect complexes on affine schemes

I'm reading a paper on algebraic stacks and in some part is stated the following:
Let $X$ be an algebraic stack and let
$P\in D_{qc}(X)$ be a perfect complex. Then, for every $x\in |X|$, there ...

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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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104 views

### Deformation theory about filtered sheaves

I am reading the paper “ Components of the stack of torsion-free sheaves of rank 2 on ruled surfaces” by C.Walter.
In the proof of Lemma 4.1(a) ,in $Filtcoh(X)$ and an object $F_1 \subset F$ the ...

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85 views

### The morphism between the moduli stack of filtered sheaves and of coherent sheaves

I am thinking about the morphism from the moduli stack of filtered coherent sheaf on $X$ to the moduli stack of coherent sheaves defined by forgetting filtrations,i.e.$Filtcoh(X)$ $\rightarrow$ $Coh(X)...

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114 views

### About morphisms of Artin stacks

I read the following question .
(Morphism of Artin stacks)
In Mr.skirvin’s answer,I found the statement “ Given surjective, representable $\phi:\mathcal{F} \to \mathcal{G}$, we obtain a surjective map ...

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**1**answer

227 views

### Can being schematic be checked on an atlas?

Let $X\to Y$ be a map between algebraic stacks, and $U\to Y$ a smooth atlas of $Y$. Suppose we know that $X\times_Y U$ is a scheme, can we show that $X\to Y$ is schematic?

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**1**answer

286 views

### K-theory for a (geometric) stack

There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...

**2**

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**1**answer

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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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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84 views

### Calculating intermediate extension on the stack of coherent sheaves of rank $1$

Let $L$ be a line bundle of degree $d$ on a curve $X$ and let $x$ be a point of $X$. I want to describe the intermediate extension of the constant sheaf from the stack of line bundles of degree $d$ to ...

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256 views

### Does the compactified Torelli map extend to a proper map of stacks?

Let $M_g^{ct}$ denote the moduli stack of compact type genus $g$ stable curves and $A_g$ the moduli stack of principally polarized $g$-dimensional abelian varieties.
Can someone provide a reference ...

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104 views

### Theta bundles on moduli space of principal G-bundles

Let $G$ be a simply connected, semi-simple affine algebraic group and $C$ be a smooth projective curve with $g \geq 3$.
Let $\mathcal{M}_G$ be a moduli stack of principal G-bundles on the curve $C$ ...

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**1**answer

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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207 views

### Notions of algebraic/differential geometry of scheme/manifolds extended to algebraic/differential stacks

Given a manifold, one can associate a stack over the category of manifolds, which is a differential geometric stack. This gives a functor $\text{Man}\rightarrow \text{D.Stacks}$. This is an embedding....

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1k views

### Visualization of an algebraic stack

As the visuallization of an algebraic stack is virtually impossible I warn about this is a soft question.
I am interested in thinking visually about algebraic stacks (also higher and derived stacks, ...

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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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**1**answer

723 views

### Generalized Behrend version for Grothendieck-Lefschetz trace formula

[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$,
$$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...

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**1**answer

174 views

### Topological realisation of a stack (explicit description)

Let $X$ be an Artin stack over $k=\mathbf{C}$. I've heard that $X$ has a topological realisation, but I've not been able to find an explicit decription.
My first guess would be: take a smooth cover $...

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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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**3**answers

947 views

### Are there prominent examples of operads in schemes?

There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...

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**1**answer

287 views

### 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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392 views

### 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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123 views

### Is this construction with stacks a blow-up?

Let $X$ be the stack of rank $1$ degree $b$ coherent sheaves $E$ with torsion of length at most 1 on an elliptic curve $C$. Let $Y$ be the stack of pairs $E^{'} \subset E$ such that $E \in X$ and $E/E^...

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111 views

### Atlas for a stack of sheaves of rank 1 with torsion

I would like to construct an atlas for the stack of sheaves E of rank 1 and degree b on an elliptic curve C such that E has torsion of length at most 1. Am I allowed to fix both the determinant L of ...

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**1**answer

170 views

### Is the stack of stable curves with no rational component algebraic?

Let $g\geq 2$ be an integer and let $\overline{\mathcal{M}}_g$ be the (smooth proper Deligne-Mumford) algebraic stack of stable curves of genus $g$.
Let $\mathcal{M}_g^{nr}$ be the substack of ...

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222 views

### 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.
...

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151 views

### A naive question about representations of group stacks

For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with ...

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159 views

### On presentations of algebraic stacks

Every algebraic stack $X$ admits a presentation as quotient stack $[U/R]$ of a groupoid in algebraic spaces
($U$ is an algebraic space of objects, $R$ is an algebraic space of arrows, and there are ...

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**2**answers

406 views

### Understand the difference between two stacks

Let us work over $\mathbb{C}$. Let $G$ be a finite group, acting on $\mathbb{A}^1$ via a character, and let $H$ be the kernel of the action.
Assume that $\mathbb{A}^1$ is the coarse moduli space of ...

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335 views

### Tangent and cotangent bundle of a smooth algebraic stack

Are there any good notion of tangent and cotangent bundle (and stacks) of a smooth algebraic stack, similar to the notion of tangent and cotangent bundles (and spaces) of smooth schemes?
I am ...

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**1**answer

131 views

### Properties inherited by the coarse moduli space

Let $\mathcal{X}$ be a regular, seperated Deligne-Mumford stack and $X$ be the coarse moduli scheme associated with $\mathcal{X}$. Then, is $X$ regular? I guess this is a basic fact but am not able to ...