# Questions tagged [algebraic-stacks]

The algebraic-stacks tag has no usage guidance.

219
questions

**6**

votes

**1**answer

228 views

### 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 ...

**5**

votes

**0**answers

138 views

### 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 $\...

**3**

votes

**1**answer

188 views

### 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,...

**2**

votes

**0**answers

176 views

### 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 ...

**2**

votes

**0**answers

151 views

### 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 $[\...

**3**

votes

**0**answers

90 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 ...

**8**

votes

**2**answers

633 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^*(-,...

**2**

votes

**0**answers

145 views

### 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_\...

**7**

votes

**0**answers

168 views

### 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 ...

**4**

votes

**0**answers

81 views

### 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 ...

**5**

votes

**1**answer

382 views

### 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 ...

**6**

votes

**2**answers

169 views

### 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) ...

**2**

votes

**0**answers

146 views

### 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 ...

**4**

votes

**1**answer

281 views

### 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 ...

**1**

vote

**0**answers

181 views

### 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/...

**3**

votes

**0**answers

145 views

### 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 ...

**2**

votes

**0**answers

56 views

### 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{...

**4**

votes

**1**answer

166 views

### 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 ...

**5**

votes

**1**answer

239 views

### 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 $...

**12**

votes

**0**answers

210 views

### 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 ...

**4**

votes

**1**answer

295 views

### 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 ...

**3**

votes

**0**answers

158 views

### $2$-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 (...

**8**

votes

**0**answers

130 views

### 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 ...

**5**

votes

**1**answer

196 views

### 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 $T$ ...

**5**

votes

**0**answers

162 views

### 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 ...

**2**

votes

**1**answer

304 views

### 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)...

**6**

votes

**1**answer

252 views

### 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 ...

**2**

votes

**0**answers

205 views

### 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)))-\...

**4**

votes

**0**answers

151 views

### 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 ...

**8**

votes

**0**answers

319 views

### 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 $\tau$-...

**3**

votes

**0**answers

138 views

### 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 ...

**8**

votes

**1**answer

394 views

### Milnor excision for algebraic stacks

Recall that a commutative square of commutative rings
$$\begin{matrix}
A&\to&B\\
\downarrow &&\downarrow\\
A^\prime&\to&B^\prime\end{matrix}$$
is called a Milnor square if the ...

**12**

votes

**0**answers

229 views

### birational geometry of moduli spaces: why work on the coarse space?

In studying the birational geometry of $\overline{\mathcal{M}}_g$, it seems standard to work with the coarse space $\overline{M}_g$ rather than the smooth stack $\overline{\mathcal{M}}_g$. Why is this?...

**6**

votes

**1**answer

188 views

### Ferrand pushouts for algebraic stacks

Given algebraic spaces $X$, $Y$, $Z$ with a finite morphism $Y \rightarrow X$ and a closed immersion $Y \hookrightarrow Z$, the pushout $P \cong X \amalg_Y Z$ exists as an algebraic space (cf. Temkin ...

**1**

vote

**0**answers

178 views

### Two definitions of cotangent complex

I have reading a paper by Professor Pridham(https://arxiv.org/abs/0905.4044v4). Page 47-48 contains a comparison of the two definitions of the cotangent complex, but there is a part I don't understand....

**3**

votes

**0**answers

104 views

### Perfect complexes on stacks are strict?

As mentioned in the comments of this question, on a quasiprojective scheme over a field, every perfect complex is globally a complex of vector bundles.
I have some question about the extension of this ...

**3**

votes

**1**answer

164 views

### Algebraic spaces in the étale topology (proof from Stacks project)

I have a question about the proof of Lemma 78.12.1 from Stacks Project. The aim of the last paragraph of the proof is to verify that the map of sheaves in the étale topology $F \to U/R$ is an ...

**1**

vote

**1**answer

205 views

### Stabilizer $G_x$ of a $k$-valued point of an algebraic Stack

An algebraic stack or Artin stack is a stack in
groupoids $\mathcal{X}$ over the étale site such that the diagonal
map of $\mathcal{X}$ is representable and there exists a smooth
surjection from (the ...

**2**

votes

**0**answers

86 views

### Infinititesimal Automorphisms intuition (algebraic stacks)

Let $F$ be a category cofibered in groupoids over category $C$. Given a morphism $x'\to x$ in $F$ lying over a morphism $A′\to A$ in $C$, there is an induced homomorphism
$\operatorname{Aut} A'(x')\to ...

**2**

votes

**0**answers

83 views

### A morphism $F$ of algebraic stacks is quasi-compact iff $|F|$ is quasi-compact

This is 5.6.3. of Laumon, Moret-Bailly's "Champs Algeriques".
Let $S$ be a scheme, $\mathscr{X, Y}$ algebraic stacks over $S$, and $F : \mathscr{X \to Y}$ a morphism.
Then $F$ is quasi-...

**3**

votes

**0**answers

143 views

### The coarse moduli space of a weighted projective line

Fix two positive integers $a$ and $b$. Consider a weighted projective line $\mathbb{P}(a,b)$ as a quotient stack
$$[(\mathbb{C}^2-\{0\})/\mathbb{C}^*]$$
where $\mathbb{C}^*$ acts on $\mathbb{C}^2-\{0\}...

**2**

votes

**2**answers

274 views

### A presentation of an algebraic stack is epi. in etale topology

Let $S$ be a scheme and $\mathscr{X}$ an Artin stack over $S$.
Let $X$ be a scheme and $P : X \to \mathscr{X}$ a representable morphism, which is smooth and surjective.
Then this $P$ is an epimorphism....

**1**

vote

**0**answers

114 views

### Properness of algebraic stacks

What is the definition of a proper algebraic Artin stack? Is there a valuative criterion?
If there is such a notion is it true that every fiber of a representable smooth morphism between two proper ...

**3**

votes

**0**answers

116 views

### Commutative group stacks and Galois cohomology

"Classically", if we consider an abelian variety $A$ over some number field $k$, we get a $Gal(\bar{k}/k)$-module $A(\bar{k})$, or equivalently a sheaf of abelian groups on the étale site $\...

**3**

votes

**0**answers

111 views

### Locally a Deligne-Mumford stack has a finite etale covering

This is a part of the proof of corollary 6.1.1 of Laumon, Moret-Bailly's "Champs Algeriques".
Let $S$ be a scheme and $\mathscr{X}$ a non empty quasi-separated Deligne-Mumford stack over $S$...

**5**

votes

**0**answers

304 views

### Definition of the cotangent complexes of Artin stacks

I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived ...

**3**

votes

**0**answers

181 views

### Etale sites for stacks

Let $X$ be an algebraic stack, let $U\to X$ be a smooth cover by an algebraic space. In this setting, we have the big étale site of $X$ (if $X$ is a stack over a scheme $S$, this is the restriction of ...

**2**

votes

**1**answer

214 views

### $f^!=f^*[d]$ for quasismooth maps?

Given a smooth map of schemes $f:X\to Y$ of relative dimension $d$, then there is a natural isomorphism $f^!\simeq f^*[d](2d)$ (in any context where the six operations are defined; see Cesinski-...

**2**

votes

**0**answers

99 views

### $\mathscr Coh_{X|S} $ is algebraic and of finite type

Let $S$ be a Noetherian scheme and $X$ a projective $S$-scheme.
Reading Laumon-Moret--Bailly's "Champs Algebriques", Theorem 4.6.2.1:
$ \mathscr Coh_{X|S} $ and $ \mathscr Fib_{X|S,r} $ are ...

**4**

votes

**0**answers

115 views

### Linear vs algebraic unipotent quotient stacks

Consider algebraic stacks of the form $\mathbb{C}^n/G$ where $G$ is a unipotent group satisfying either
Type 1: $G$ acts on $\mathbb{C}^n$ via affine linear transformations
Type 2: $G$ acts on $\...