# Questions tagged [stacks]

In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.

500
questions

3
votes

1
answer

126
views

### Extending Tannakian "dictionary" to gerbes

The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".
Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...

8
votes

0
answers

289
views

### Do automorphisms actually prevent the formation of fine moduli spaces?

I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...

3
votes

0
answers

126
views

### Is the classifying stack of an abelian variety separated?

If $G/k$ is an algebraic group, a necessary condition for $BG$ to be separated over $k$ is that $G$ is proper over $k$. Assuming that $G$ is smooth and connected we are reduced to the case of an ...

3
votes

0
answers

109
views

### Why this morphism of stacks is an isomorphism?

Let $C$ be a site and $p : F \mapsto C$ $p' : F' \mapsto C$ two categories fibred in groupoids over $C$ which are stacks (see, e.g., the definition here https://stacks.math.columbia.edu/tag/0268). For ...

2
votes

0
answers

55
views

### Stack of smooth fiber bundles with fiber $F$

I'd like to premise that while I know the definition of (differentiable) stack, I'm not really into the language of schemes so my understanding of what is a moduli stack is pretty concrete and ...

3
votes

0
answers

184
views

### Category of sheaves of vector spaces on BG

Let $G$ be an affine group scheme over $\mathbb{C}$. I am interested in understanding the differences between different notions of sheaves on the stack $pt/G = BG$. For any algebraic stack $X$ one can ...

5
votes

0
answers

291
views

### What, precisely, is a stratification of a stack?

I'm currently working on a structure result for a certain spectral moduli problem, and I've been running into the problem of having to define what, precisely, is meant by the term "stratification&...

7
votes

0
answers

114
views

### Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid

This questions is about the distinction between:
Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...

1
vote

0
answers

45
views

### 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 ...

5
votes

0
answers

152
views

### Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks

I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...

4
votes

0
answers

131
views

### Nice proof that de Rham complex computes Lie algebra cohomology?

If $G$ is a nice enough group acting on a nice enough space $X$, then the relative de Rham complex
$$\Omega^\bullet_{X/(X/G)}\ \simeq\ \mathcal{O}_X\otimes\text{Sym}\,\mathfrak{g}^*[-1]$$
is given by (...

3
votes

0
answers

141
views

### Relationship between $\infty$-categories of ind-coherent sheaves on the base and total space

My question is vaguely as follows, let $G\to E\to X$ be a principal $\infty$-bundle for some group object $G$ in a category of ($\infty$-)stacks. Can we recover the category $\operatorname{IndCoh}(E)$ ...

2
votes

0
answers

236
views

### Finite generation of stack cohomology

Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra?
For instance, $\text{H}^*(\text{B}\mathbf{G}...

2
votes

0
answers

91
views

### 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}: \...

2
votes

0
answers

165
views

### Dualizing sheaf for classifying stack and duality

For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...

1
vote

0
answers

176
views

### Moduli stack of l-adic sheaves?

Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors.
Let $\ell$ be a prime ...

9
votes

0
answers

261
views

### 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 (...

2
votes

0
answers

145
views

### Infinitesimal criteria for unramified morphism on stacks

In an Artin stack, the tangent complex is fundamentally a derived object, often viewed as a 2-term complex via some quotient presentation, as discussed in Raskin's note here, for example. In ...

1
vote

0
answers

133
views

### What is bad when stabilizers are non-reductive in moduli stacks?

Here is J. Alper's definition of good moduli spaces.
Consider in characteristic zero. Then we see that the classifying stack of any non-reductive group $H$ does not have a good moduli space. In ...

6
votes

1
answer

365
views

### How to differentiate natural transformations?

Let $G_{\bullet} = (G_1 \rightrightarrows G_0)$ and $H_{\bullet} = (H_1 \rightrightarrows H_0)$ be Lie groupoids and $\varphi_\bullet, \psi_\bullet : G_\bullet \to H_\bullet$ be Lie groupoid morphisms,...

1
vote

0
answers

291
views

### G-local systems via the classifying stack BG

First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...

11
votes

1
answer

773
views

### Gerbes over finite fields

Let $k$ be a field with algebraic closure $\bar{k}$.
Recall that a gerbe over $k$ is an algebraic stack $\mathcal{G}$ over $k$ such that the groupoid $\mathcal{G}(\bar{k})$ is connected. We say that $\...

5
votes

1
answer

216
views

### Morphisms of fibered categories which are compatible with the chosen cleavages

Let $\pi \colon F \rightarrow C$ and $\pi' \colon F' \rightarrow C$ be two fibered categories over the category $C$. A morphism from $\pi \colon F \rightarrow C$ to $\pi' \colon F' \rightarrow C$ is a ...

1
vote

1
answer

240
views

### Examples when algebraic 1-stack = derived enhancement?

Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?
Let me take an example from notes of Bertrand Toen, page 41 of https:/...

6
votes

1
answer

386
views

### Anafunctors vs the plus construction

Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations
$$G(M) := \text{...

0
votes

1
answer

112
views

### Covering a stack by an open substack that contains all points of finite type

Let $X = \text{Spec}A$ be an affine scheme. It is well known that if $U$ is an open subset which contains $\text{SpecMax}A$, then $U\supseteq X$. The previous statement generalizes to arbitrary ...

1
vote

0
answers

75
views

### Function vanishing on the image of a morphism of algebraic stacks

Let $f: X\longrightarrow Y$ be a morphism of algebraic stacks, and assume I know the Krull dimension of the ring of global functions on $Y$ to be $n$. Assume that the stack $X$ has "stacky" ...

6
votes

0
answers

232
views

### Non-invertible map between principal bundles only locally trivial in a supercanonical Grothendieck topology?

It is a truth universally acknowledged that a map between principal bundles must be in want of an inverse.
However, the construction of said inverse in the context of a more general site $(S,J)$ ...

10
votes

1
answer

372
views

### 2-completeness of stacks

I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...

23
votes

1
answer

3k
views

### Is there a ring stacky approach to $\ell$-adic or rigid cohomology?

Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...

5
votes

1
answer

568
views

### Proving Zariski descent

I want to understand why the functor $\mathscr{D}$
sending an affine scheme to its associated derived
$\infty$-category satisfies Zariski descent. My understanding
is that one has to show that given a ...

2
votes

0
answers

186
views

### Picard group of a DM-stack

I wonder if the followings are true. Let $k$ be a field, $G/k$ is a finite constant group scheme acting on an affine $k$-scheme $\mathrm{Spec}(A)$. Then a line bundle on the quotient DM stack $[\...

1
vote

0
answers

151
views

### Moduli of morphisms between varieties

Let $\mathcal{M}, \mathcal{N}$ be two "well-behaved" (i.e. representable by an algebraic stack parametrising flat, proper, surjective, finitely presented morphisms with geometric fibres ...

5
votes

0
answers

88
views

### Stacks v.s. Stratifolds?

Both stacks and stratifolds can model spaces with singularities (say, singular quotient space for example). In my little experience, stacks are widely used in moduli problems and stratifolds are often ...

6
votes

0
answers

211
views

### Local structure of smooth morphisms of stacks

Let $\varphi:X \to Y$ be a smooth morphism of schemes.
There is a well-known local structure theorem: Zariski locally $\varphi$ is given by the composition of an etale morphism and an affine space (...

3
votes

0
answers

244
views

### When a stack quotient coincides with GIT quotient?

Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive.
Question: Is it true that when $G/H$ is open in its affine ...

1
vote

0
answers

115
views

### Computing the equivariant Chern character

Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ ...

2
votes

1
answer

294
views

### Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves

I am reading this paper https://arxiv.org/abs/1608.04797
Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh_P$ be the ...

1
vote

0
answers

134
views

### The stack $\operatorname{GL}_2/B$

Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...

2
votes

0
answers

120
views

### Moduli stack of doubly periodic complexes?

Let $\mathcal{A}$ be an abelian category.
In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...

7
votes

1
answer

353
views

### Faithfully flat descent in complex analytic geometry

A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "...

5
votes

1
answer

339
views

### How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point

I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
Setting
Let $\mathcal{M}_{1, 1, k}$ ...

2
votes

0
answers

394
views

### Understanding the universal family $\mathcal{M}_{g,1}\to\mathcal{M}_g$

I am trying to understand why the morphism of prestacks $\mathcal{M}_{g,1}\to\mathcal{M}_g$ is the "universal family". Here $\mathcal{M}_g$ is the moduli stack of curves of genus $g$, its ...

6
votes

1
answer

463
views

### Completion of the classifying stack $BG$ at a point

With the classifying stack $BG$ I have come across "the formal completion of $BG$ at point", which is denoted $\widehat{BG}$, for instance on page 7 of https://arxiv.org/pdf/1703.08578.pdf, ...

3
votes

0
answers

162
views

### Linear deformations of a morphism between stacks

Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$?
In ...

1
vote

0
answers

311
views

### Deformation theory of stacks and the tangent complex

On a smooth stack X one can construct the two-term tangent complex $T_X \in D(X)$, as in Sam Raskin's notes (https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf).
I ...

6
votes

1
answer

375
views

### Fibers of the coarse moduli space map

Let $\mathcal{X}$ be a Deligne-Mumford stack over a field $k$ which admits a coarse scheme $c : \mathcal{X}\rightarrow X$. This will be the case if $\mathcal{X}$ is separated and locally of finite ...

1
vote

0
answers

128
views

### Extending Grothendieck topology from analytic manifolds to spaces?

Let $k\text{-Man}$ denote the Euclidean site of $k$-analytic manifolds where $k=\mathbb{R}, \mathbb{C}$. In words, $k\text{-Man}$ is the usual category of real/complex analytic manifolds considered ...

0
votes

0
answers

185
views

### What happens if you apply $\operatorname{QCoh}$ to $(X,\underline{\mathbb{C}}_X)$?

$\DeclareMathOperator\QCoh{QCoh}
\newcommand\numC{\mathbb{C}}
\newcommand\numQ{\mathbb{Q}}$Let $X$ be a topological space (possibly a scheme). What happens if you apply $\QCoh$ to the prestack $(X,\...

11
votes

0
answers

367
views

### Moduli stacks of algebraic surfaces—obstructions to existence?

The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...