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5
votes
0answers
179 views

Does every stack with a connection admit an atlas with a connection?

Dear all, Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...
4
votes
2answers
593 views

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 & ...
6
votes
1answer
352 views

Why are sheaves not preserved in this case?

Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi_{\mathscr{X}}:\int_{C} \mathscr{X}\to C$$ denote the associated fibered ...
7
votes
0answers
353 views

Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem

I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...
7
votes
0answers
182 views

Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...
2
votes
0answers
369 views

what's the coherent sheaf and sheaf cohomology on Deligne-Mumford stack ?

hello, everyone, I am reading Deligne-Mumford's paper, and the section 4 of this paper, they given some properties of Deligne-Mumford stack without proof. I can't understand the coherent sheaf and ...
2
votes
0answers
309 views

Is the category of quasi-coherent sheaves on a concentrated stack locally finitely presentable?

Let's call an Artin stack $X$ concentrated iff it is quasi-compact and quasi-separated (the latter usually being included in the definition of an Artin stack). The category of quasi-coherent sheaves ...
2
votes
1answer
282 views

Classification of principal G-bundles over a differentiable stack

According to "Notes on differentiable stacks" by Heinloth, the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13) (Here $G$ is a Lie group.) My questions are: (1) What ...
11
votes
0answers
318 views

Atiyah-Bott from Beauville-Laszlo

This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they ...
7
votes
1answer
458 views

To what extent does Poincare duality hold on moduli stacks?

Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the ...
4
votes
0answers
179 views

Properties of the Zariski-Riemann topology on the set of valuations

One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology. ...
7
votes
3answers
633 views

Reference for the derived category of $X$, $[X/G]$ and $X/G$

I'm trying to learn about derived categories of algebraic stacks. To be honest, as of now, I don't need anything fancy nor deep. In my setup I have a scheme $X$ (well, a smooth and projective variety ...
2
votes
1answer
518 views

Does Hom(X,Spec R) = Hom(R, O(X)) hold for algebraic stacks?

For an affine scheme $Spec R$ and a scheme $X$ we know that $Hom(X,Spec R) = Hom(R,\Gamma(X,\mathcal{O}_X))$. Does it still hold when we replace $X$ with an algebraic stack? My guess is yes as ...
0
votes
0answers
145 views

Terminology for a partition of unity for an étale groupoid

I would like to ask about terminology for a partition of unity for an étale groupoid. I am reading the lecture notes "Cohomology of Stacks" by Prof. Behrend. A partition of unity is defined in ...
5
votes
1answer
1k views

surjective morphism of schemes or epimorphism of sheaves?

I have a technical question coming from reading Toen's master course on stacks. If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
3
votes
0answers
206 views

properness of stack

Hi, assume we have an algebraic stack $A$ over $Sch(\mathbb{Z})$ which is quasi-compact and with separated diagonal. Assume that I have a stack $B$ which is obtained by rigidifying $A$ by a subgroup ...
7
votes
2answers
855 views

Are non-algebraic stacks useful in algebraic geometry?

The title is a bit vague. What I want to know is if there is any geometric application of non-algebraic stacks. I know e.g. the category of coherent sheaves is an example. But I want to ask if people ...
3
votes
0answers
200 views

Derived pullback of quasi-coherent complexes between algebraic stacks

Let $X$ and $Y$ be Artin algebraic stacks, and $f:X\to Y$ a morphism. I am interested in a pullback morphism $Lf^\ast : D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$. In his paper Sheaves on Artin stacks, ...
2
votes
1answer
274 views

can one define the pullback between stacks of coherent sheaves for non-flat morphisms?

Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}_X, \mathcal{M}_Y$. Does one have for free an induced pullback ...
3
votes
2answers
378 views

Algebraic stacks as (etale) groupoid alg.spaces/schemes

Assume given an algebraic stack() $\mathcal{X}$ with presentation $X_0 \to \mathcal{X}$, and the corresponding groupoid $X = (X_0\times_\mathcal{X} X_0 \rightrightarrows X_0)$ in algebraic spaces (or ...
5
votes
1answer
444 views

Query on comment in Deligne-Mumford (1969)

In Deligne and Mumford's famous 1969 paper, The irreducibility of the space of curves of given genus, definition 4.6 (that of algebraic stacks) has the following footnote: This definition is the ...
1
vote
2answers
458 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 ...
4
votes
0answers
370 views

Stack cohomology vs cohomology of the quotient

Let $X$ be an affine scheme over an algebraically closed field $K$ of positive characteristic, $G$ a finite group acting on $X$, $[X/G]$ the quotient stack and $p:[X/G]\to X/G$ the natural map of ...
5
votes
2answers
412 views

(Sh,Sh-map) represents the category of sheaves on a stack.

I'm trying to understand the following theorem, but I don't think I'm reading it correctly. Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid of ...
15
votes
3answers
2k views

Is every algebraic space the quotient of a scheme by a finite group?

In this MO question it is claimed that a catchphrase for "algebraic spaces" could be that they are "the result of looking at the orbit space of the action of a finite group on a scheme". Hence my ...
17
votes
2answers
2k views

The different types of stacks

This question is very naive, but it will help me a lot in getting in to the vast literature about stacks. The question is this: there are many kinds of stacks (algebraic spaces, DM, algebraic stacks, ...
8
votes
4answers
709 views

Adjunctions form a stack

Let $C$ be a base category, $F,G$ be two categories fibered over $C$ and $F \to G$ be a morphism. The following criterion is used very often: If all the fiber functors $F_U \to G_U$ ($U \in C$) are ...
6
votes
1answer
658 views

Basic questions about stacks 2

I have again three basic questions about stacks. 1) When we consider categories fibered in groupoids, do we always mean small or essentially small groupoids? Especially I want to know if algebraic ...
2
votes
1answer
284 views

Irreducibility of quotient stacks.

Let $[X/G]$ be a quotient stack such that $X$ is irreducible and $G$ acts trivially on $X$ (I am just adding automorphisms to every point). Under which hypothesis is $[X/G]$ irreducible as an Artin ...
11
votes
1answer
927 views

Qcoh(-) algebraic stack?

The $2$-functor $\text{Qcoh} : \text{Sch}^{op} \to \text{Cat}$, which sends a scheme to its category of quasi-coherent modules, is a stack by Descent Theory. Is it actually an algebraic stack? If not, ...
9
votes
1answer
610 views

“Approximating” $BGL(1)$ by projective spaces

Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant ...
5
votes
2answers
459 views

Automorphism groups and etale topological stacks

Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a ...
2
votes
0answers
508 views

Coarse Moduli space of proper Deligne-Mumford stacks

Let F be a finite type proper Deligne-Mumford Stack over a perfect field. Is it true that the coarse moduli space of F is proper?
5
votes
1answer
604 views

What is the stalk of a stack?

When we study sheaves of sets (on a space X or a site C) we are often interested in the stalks of the sheaf (at either a point $p:1\to X$ or a left exact, cover-preserving functor $a:C\to Sets$). I ...
10
votes
0answers
438 views

Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles. I will ...
4
votes
1answer
346 views

Ramification formula for orbifolds

It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this ...
10
votes
2answers
789 views

Basic questions about stacks

I'm trying to understand some basics of stacks in algebraic geometry and have three questions: 1) As far as I understand, the moduli stack of vector bundles over a scheme $X$ is a replacement for the ...
13
votes
1answer
497 views

Does a degeneration always have a larger-dimensional automorphism group?

Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an isotrivial degeneration of $\xi$). ...
7
votes
1answer
673 views

Exactly how is 'the diagonal is representable' used for algebraic stacks…

...apart from stating properties of $(s,t):X_1 \to X_0\times X_0$ for the a presenting algebraic groupoid $X_1 \rightrightarrows X_0$? Once we know that given a stack $\mathcal{X}$ we have a smooth ...
5
votes
1answer
365 views

Categories of descent data

Let us work over the etale site $\mbox{Aff}/S$ (for the sake of definiteness) for some fixed base scheme $S$, where the covers are jointly surjective etale maps $\{ U_i \rightarrow U\}_{i\in I}$ (and ...
5
votes
2answers
773 views

What about stacks of categories in algebraic geometry? II

I've made this a new question, rather than expanding the first one. Torsten gives a good answer, and it partially illustrates in practice the 'second approach' I outlined in my other question. (You ...
10
votes
1answer
466 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 ...
19
votes
1answer
1k views

What about stacks of categories in algebraic geometry?

Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...
11
votes
0answers
346 views

Is there a non-quotient stack with affine stabilizers whose good moduli space is a geometric point?

Definitions: One says that a map $\pi\colon\mathcal X\to X$ from an algebraic stack to an algebraic space is a good moduli space if $\pi$ is cohomologically affine and universal for maps to schemes. ...
5
votes
0answers
130 views

Maximal algebraic sub-groupoids

By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in the latter case are ...
2
votes
1answer
385 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 ...
8
votes
1answer
777 views

Why is lack of functoriality of the Lisse-Etale topology specific to the Lisse-Etale topology?

I'm trying to follow the explanation given in Olsson's "Sheaves on Artin stacks" for the lack of functoriality for lisse-étale topology: Let $f:Y \to X$ be a morphism of algebraic stacks. The functor ...
1
vote
0answers
142 views

$G_m$-cohomology of a motif (that corresponds to a stack?)

As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety I am in the following situation: $G$ is an algerbraic group, and X is a smooth ...
5
votes
2answers
468 views

Is there a “geometric” language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions. Here, ...
3
votes
2answers
474 views

Sheaves on stacks and interesting functors

Let $G$ be a finite group and $H \subset G$ a normal subgroup. Consider $G$, $H$, and $X=G/H$ as affine algebraic groups over some algebraically closed base field $k$. I hear that there is an ...