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1
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0answers
151 views

Irreducibility of monodromy of eigenspaces of families of cyclic coverings

In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in ...
7
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1answer
323 views

Chern classes of vector bundles on a stack

Is there literature about chern classes of vector bundles on DM-stacks? I had a look at a lot of different papers about intersection theory on stacks and related stuff and this seems to be known, but ...
1
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0answers
168 views

stack quotient question

Hi, I have the following question: let $k$ a field with $char(k)= p>0$, which we can assume to be perfect, $W(k)$ the ring of Witt vector, and $a,b$ positive integers. Consider the ring ...
3
votes
1answer
235 views

Representability of Hom-sheaves of various moduli spaces

(May be a poor title, happy to update) Recall that for a stack $\mathcal{X} \to Sch$ on schemes (e.g. fppf site) and a pair of morphisms $x,y\colon U\to \mathcal{X}$ ($U$ a representable stack) there ...
15
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3answers
2k views

Stacks in modern number theory/arithmetic geometry

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
3
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0answers
201 views

Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks

Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$ Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that ...
6
votes
2answers
293 views

how to construct a $C^\infty$ stack from a holomorphic stack

Given a complex manifold, you can `weaken' its structure to give a smooth manifold. Is there an analogous construction that constructs a stack over the category of smooth manifolds from a stack over ...
5
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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
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2answers
600 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
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0answers
354 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
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0answers
186 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
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0answers
371 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
313 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
284 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
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0answers
328 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
463 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
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0answers
181 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
666 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
521 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
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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
210 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
866 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
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0answers
207 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
277 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
387 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
449 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
476 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
376 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
415 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
710 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
668 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
290 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
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1answer
968 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
460 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
514 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?
6
votes
1answer
614 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
439 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
358 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
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2answers
794 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
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1answer
502 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
685 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
370 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
788 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
475 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 ...
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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 ...