# Questions tagged [algebraic-stacks]

The algebraic-stacks tag has no usage guidance.

237
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### 2-shifted 2-form on the classifying stack 𝐵𝐺

Let $G$ be a reductive group. A $2$-shifted $2$-form on the classifying stack $BG$ is by definition a a morphism of quasi-coherent complexes
\begin{equation}
\mathcal O_{BG}\rightarrow (\wedge^2 \...

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### $\mathbb G_{\mathrm{m}}$-gerbes are to (derived) Azumaya algebras as $G$-gerbes are to …?

Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection ...

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### Base change and quotient stacks

Let $X$ be an algebraic space over a scheme $S$, let $G$ be a smooth group scheme over $S$, acting on $X$, and let $T\to S$ be a morphism of schemes. Is it true that there is an isomorphism
$$[X/G]\...

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### In what sense is an orbifold a DM stack?

My advisor mentioned in passing that orbifolds are Deligne-Mumford stacks, and I'd like to know in which sense this is true. The only reference I can find is this article (https://arxiv.org/abs/0806....

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### Representability of blow-up of stacks

Suppose $\mathcal{Z}\subseteq\mathcal{X}$ is a closed immersion of stacks. Is the blow-up morphism $p:Bl_{\mathcal{Z}}(\mathcal{X})\to\mathcal{X}$ representable? More precisely, if $C\to\mathcal{X}$ ...

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### Quotient of a quotient stack: interesting examples?

Let $X$ be a scheme acted on by an algebraic group $G$. Also, let $H$ be an algebraic group acting on the quotient stack $X/G$, for the definition of "act", see Romagny - Group Actions on ...

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### Geometric quotients of DM stacks by group actions

Let $G$ be a finite group acting on a DM stack $X$ and $Y$ the quotient stack. I.e., $Y \to BG$ has fiber $X$. Is there a geometric quotient $X/G$ in some sense? I want automorphism groups of points ...

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### Is there a degeneration formula for Gromov-Witten K-theoretic invariants?

By Gromov-Witten K-theoretic invariants (call them KGW) I mean the invariants defined by Givental and Lee.
I expect the formula expresses the KGW of the generic fiber of a given degeneration in terms ...

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### Square root of a line bundle up to a finite surjective morphism

Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$.
The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^...

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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 ...

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### 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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256
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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....

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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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### Finiteness result for higher direct image of $\ell$-adic sheaves

Let $f:X\to Y$ be a representable map of finite type (or is finite dimensional enough?) Artin stacks, whose fibres (which are schemes) have dimension at most $n$. Then is it true that $R^qf_*\mathbf{...

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### Chevalley complex and $\text{BG}$

For a long time I've been under the impression that the Chevalley complex $\text{CE}(\mathfrak{g})$ of a semisimple (maybe can weaken this) Lie algebra $\mathfrak{g}$ can be extracted from the ...

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### Measuring non-separatedness of algebraic stacks

Let $X$ be an algebraic stack of finite type over a field $k$, and let $U\subset X$ be an open dense separated sub-scheme of $X$. Let $D=\text{Spec}~ k[[t]], D^*=\text{Spec}~ k((t))$.
Fix a map $f:D^*\...

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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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### Stacky points detect nilpotent cohomology

Take a finite type scheme $X/\mathbf{C}$ acted on by a reductive group $G$. Then $X/G$ is an Artin stack. Let $\alpha\in H^*(X/G)$ be a rational cohomology class on it.
Question: Is it true that $\...

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### Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)

In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer):
''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...

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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 ...

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### The classifying stack of $\operatorname{PGL}(2)$ and the moduli space of genus zero curves

$\DeclareMathOperator\PGL{PGL}$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 $[\...

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### 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 ...

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### 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^*(-,...

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### Generalizations of Artin–Verdier duality?

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_\...

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### Moduli stacks and representability of diagonal by schemes

The answer to my question might very well be standard, but I have had trouble finding the right keywords to search for it, so I apologize if this is something well-known to the experts.
I am learning ...

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### Classifying twists for a general moduli problem

Suppose $\mathfrak X$ is a (Deligne-Mumford/Artin/...) stack and denote by $|\mathfrak X(k)|$ the set of isomorphism classes of it's $k$-valued point. Can we say something in general about the fibers ...

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### About an argument in Olsson's book

The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson.
I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end.
It seems that there might ...

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### Relation between finite dimensional representations of an affine group scheme and quasicoherent sheaves on the classifying stack

Let $G$ be an affine group scheme over a field $k$ of characteristic zero.
I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) ...

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### Are universal geometric equivalences of DM stacks affine?

Let $f:X\to Y$ be a map of Deligne-Mumford stacks. Let's say that the map $f$ is a geometric equivalence if the induced map on small étale topoi is a geometric equivalence. Moreover, let's say that ...

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### A de Rham space for meromorphic connections?

To any space $X$ you can associate its de Rham space $X_{dR}$. Vector bundles on $X_{dR}$ are the same thing as vector bundles on $X$ with a flat connection.
Can anything like this be said for ...

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### Is there an intrinsic Gauss map?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$This may all be well known, too vague, or stupid; my apologies.
The Gauss map is defined by embedding your $k$-dimensional smooth scheme/...

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### Sheaf $\operatorname{Isom}(x,y)$ isomorphic to fibered product $U \times_{(x,y),X \times X, \Delta} X$

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Isom{Isom}$Let $S$ be a scheme and $C$ be the category $(Sch/S)$. Let moreover $p:X \to C$ be an algebraic stack over $C$. Consider an arbitrary ...

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### Map from the stack of coherent sheaves on a curve to the Grothendieck group

Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map
$$
Coh(X)\rightarrow Pic(X)\times \mathbf{...

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### Do quotient stacks help classify the orbits of group actions on varieties?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting ...

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### Construction of an atlas for the moduli stack $\mathcal{Bun}_X^{n,d}$ in F. Neumann's 'Algebraic Stacks and Moduli of Vector Bundles'

I'm reading Frank Neumann's "Algebraic Stacks and Moduli of
Vector Bundles" and have some problems to understand
a construction from the proof of:
Theorem 2.67. (page 81) The moduli stack $...

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### Do connected algebraic stacks have a smooth cover by a connected scheme?

An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there ...

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### Are universally submersive morphisms of stacks universal descent morphisms for relative étale stacks?

Recall that a map of schemes, algebraic spaces, stacks, etc is called submersive if the associated map on underlying topological spaces is a quotient map. Recall moreover that a map is called ...

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### $2$-vector spaces and algebraic $2$-stacks

I am thinking about higher Artin stacks in the sense of Simpson, concretely I would like to calculate the dimension and compare these two cases:
$\mathfrak{X}_{1}=$ Higher linear stack classifying (...

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### Closed immersion → Pro-open immersion factorization for residual gerbes

Let $X$ be a quasi-separated algebraic stack. Then it is a theorem of Rydh that every point $x$ in $X$ admits a residual gerbe. More or less, the construction proceeds by first taking the closure ...

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### When quotient stacks (for nonsmooth group) are algebraic and related questions

Let $k$ be a field. Consider a group $k$-scheme $G$ and let $X$ be a $k$-scheme equipped with an action of $G$. Then one can define the quotient stack $[X/G]$. Objects of $[X/G]$ over $k$-scheme $T$ ...

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### Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...

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### K/G-theory of affine bundles

Setting: $f : C \to D$ is a morphism of Artin stacks over $X$ which is a torsor for a vector bundle $T \to X$: étale-locally in $X$, we have $C \simeq D \times_X T$. I want to conclude that $f^*: G(D)...

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### Zariski's main theorem for non-representable morphisms?

Let $f:\mathcal{X}\to \mathcal{Y}$ be a separated quasi-finite map of qcqs Deligne-Mumford stacks. Is there a version of Zariski's main theorem that makes sense in this context? Rydh proved a ...

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### Dimension of the moduli stack of vector bundles over a curve

Let $Vect_{n}(C)$ the moduli stack of vector bundles $V$ of rank $n$ over a smooth curve $C$ of genus $g$. It is well known that $Vect_{n}(C)$ is a smooth stack of dimension $\dim(H^{0}(C,End(V)))-\...

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### Gysin map and $B\mathbf{G}_m$, confusion

Write $\text{Sh}(X)$ for the triangulated/stable $\infty$ category of $\ell$-adic sheaves on $X$, and $k\in\text{Sh}(X)$ fo the unit object. In playing around with $\text{Sh}(B\mathbf{G}_m)$ I've ...

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### Geometric stacks, groupoids and étendues

If $(C, \tau)$ is a site with pullbacks and $\tau$ subcanonical, it is well known that these things are essentially equivalent:
Groupoids $s,t: U_1 \to U_0$ where $s,t$ are covering for the $\tau$-...

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### Dimension of derived Artin stacks and perfect complexes

I am interested in the concept of dimension of derived and $n$-Artin stacks. Take for example the definition 4.10 of From HAG to DAG: derived moduli stacks. in which they define the dimension of a ...