Skip to main content

Questions tagged [stacks]

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

Filter by
Sorted by
Tagged with
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 ...
bsbb4's user avatar
  • 323
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 ...
Coherent Sheaf's user avatar
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 ...
Aitor Iribar Lopez's user avatar
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 ...
Analyse300's user avatar
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 ...
Kandinskij's user avatar
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 ...
arczn's user avatar
  • 53
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&...
Doron Grossman-Naples's user avatar
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, $G_1 \;{}_s\!\times_t G_1$, ...
Konrad Waldorf's user avatar
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 ...
kindasorta's user avatar
  • 2,103
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 $\mathscr{X}$ ...
Hajime_Saito's user avatar
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 (...
Pulcinella's user avatar
  • 5,565
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 $\operatorname{IndCoh}(E)$ ...
Grisha Taroyan's user avatar
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}...
Pulcinella's user avatar
  • 5,565
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}: \...
Matthias's user avatar
  • 203
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 ...
E. KOW's user avatar
  • 752
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 ...
user577413's user avatar
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 (...
Tim Santens's user avatar
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 ...
C.D.'s user avatar
  • 565
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 ...
Display Name's user avatar
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,...
Felix Lungu's user avatar
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 $G$-...
Emerson Hemley's user avatar
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 $\...
Daniel Loughran's user avatar
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 ...
Adittya Chaudhuri's user avatar
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:/...
Robert Hanson's user avatar
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{...
Connor Grady's user avatar
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 ...
kindasorta's user avatar
  • 2,103
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" ...
kindasorta's user avatar
  • 2,103
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 $(S,J)$ ...
David Roberts's user avatar
  • 34.8k
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 ...
Nico's user avatar
  • 775
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 ...
Gabriel's user avatar
  • 1,139
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 ...
user141099's user avatar
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 $[\...
mhahthhh's user avatar
  • 435
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 ...
Matthias's user avatar
  • 203
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 ...
Ruizhi liu's user avatar
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 (...
Daniel Loughran's user avatar
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 ...
Ruotao Yang's user avatar
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 $D^b([X/G]) \simeq D^b(X)^G$ ...
alg_et_geom's user avatar
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 ...
angry_math_person's user avatar
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 $\...
kindasorta's user avatar
  • 2,103
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 ...
Pulcinella's user avatar
  • 5,565
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 "...
G. Gallego's user avatar
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 $\mathcal{M}_{1, 1, k}$ ...
PIELEO13's user avatar
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 ...
Sergey Guminov's user avatar
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, ...
Robert Hanson's user avatar
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 ...
Robert Hanson's user avatar
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 ...
Robert Hanson's user avatar
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 ...
stupid_question_bot's user avatar
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 ...
Andy Sanders's user avatar
  • 2,890
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,\...
Gaussler's user avatar
  • 295
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 ...
Pulcinella's user avatar
  • 5,565

1
2 3 4 5
10