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# Questions tagged [algebraic-stacks]

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### etale fundamental group of global quotient algebraic stacks

I am reading about fundamental group of algebraic stacks, and in my opinion, the class of quotient stacks is an important one in algebraic stacks. However, I am not clear about how to compute ...
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### Defining log prestacks (and their structures)

It's possible to define log schemes, and Olsson's thesis defines log stacks. Is there a definition of log prestack in the literature? Is it easy to extend Gaitsgory-Rozenblyum type results (...
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### Residual gerbes and coarse moduli space of a DM stack

Let $X$ be a nice DM stack (Noetherian, separated), and $X_0$ its coarse moduli space which exist by the Keel-Mori theorem. (I like the exposition in D. Rydh. “Existence and properties of geometric ...
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### 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 (...
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### Decomposition of vector bundles on the inertia stack of a DM stack

Let $X$ be a tame DM stack over $\mathbb{C}.$ Let $IX$ denote the inertia stack of $X.$ Let $K(IX)$ denote the Grothendieck group of vector bundles on $IX.$ By the discussion on page 20 of Toen's ...
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### Connected components of inertia stacks

Let $k$ be a field. Let $X$ be a connected tame DM stack over $k.$ Let $IX$ be the inertia stack of $X.$ Then $IX$ is a disjoint union of connected components. Is this always a finite union? If not, ...
1 vote
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### Smoothness of inertia stacks

Let $k$ be a field of characteristic zero. Let $X$ be a smooth DM stack over $k.$ Is the inertia stack $IX$ always smooth over $k$? I believe this is true, but cannot find a proof in the literature. I ...
1 vote
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### Cohomology of quotient stack

Let $X$ be an algebraic variety over $\mathbb{C}$ (the ground field is not important but this makes things easier I think) and $G$ an algebraic group acting over it. Let's say we know that there's a ...
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### Moduli stack of quiver representations

Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT ...
385 views

### Residue field of point on an algebraic stack

$\DeclareMathOperator{\Spec}{Spec}$ Let $X$ be an algebraic stack. Is there is a well-defined notion of the residue field of a point $x \in |X|$? Attempts: Recall that a point on a stack is an ...
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### Character stack and character variety

Let $\Sigma$ be a Riemann surface of genus $g$. We can consider two different type of objects associated to it parametrising representations of its fundamental groups. On one side we have the ...
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### Artin's "Versal Deformations and Algebraic stacks": Question concerning proof of Theorem 3.3

I have been reading Artin's paper titled "Versal deformations and algebraic stacks" and am a bit confused about a statement he makes in the proof of Theorem 3.3, in the first few lines of pg....
1 vote
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### Generic stabiliser of moduli space of vector bundles on a curve

Let $C$ be a curve of genus $g >1$. Fix a line bundle $L$ on $C$. Let $Bun_{n,L}$ be the moduli stack of vector bundles of rank $n$ on $C$ with determinant isomorphic to $L$. I have a few questions ...
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### Criterion for relative Deligne-Mumfordness?

A quotient stack $X/G$ of a scheme by a smooth algebraic group is Deligne-Mumford if and only if (approximately) the stabiliser groups of points are finite (for the precise statement see corollary 8.4....
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### Irreducible components of an algebraic stack

Let $\mathcal{X}$ be an algebraic stack of finite type over a (separably closed) field $k$. Let's say that $\mathcal{X}$ has finite dimension $d \in \mathbb{Z}$. Is it still true that the number of ...
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### Relation between stacky curves and "M-curves"

A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
$\DeclareMathOperator\PGL{PGL}$The classifying stack of $\PGL(2)$ is the stack quotient $[\operatorname{Spec} k/\PGL(2)]$ where $\PGL(2)$ acts trivially on $\operatorname{Spec} k$. Since $[\... 3 votes 0 answers 124 views ### Automorphism of a stack morphism Let$X$be an algebraic stack and let$f: S \to X$be a smooth covering of$X$by a scheme$S$. Motivation: Forgetting about stacks for a moment and going back to covering spaces: Given a covering map ... 9 votes 2 answers 702 views ### Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition? I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning. Let$M$be a manifold, and consider the presheaf$C^*(-,...
Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than \$Spec\ O_\...