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### 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 ...

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### 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 ...

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### 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 ...

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198 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 ...

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291 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 ...

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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 ...

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597 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 & ...

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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 ...

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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 ...

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### 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 ...

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370 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 ...

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311 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 ...

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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 ...

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321 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 ...

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461 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 ...

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### 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.
...

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657 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 ...

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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 ...

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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 ...

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### 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 ...

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207 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 ...

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### 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 ...

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### 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, ...

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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 ...

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### 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 ...

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447 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 ...

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461 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 ...

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### 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 ...

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### (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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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287 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 ...

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### 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, ...

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### “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 ...

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### 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 ...

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### 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?

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609 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 ...

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### 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 ...

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352 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 ...

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### 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 ...

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### 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 $\xi$). ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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. ...

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### 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 ...