Questions tagged [algebraic-stacks]

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Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?

I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$. Recently I was pointed to Katz and Mazur's book, ...
David Roberts's user avatar
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3 votes
0 answers
71 views

References for orbifold curves

I am looking for a good reference (if there is any) for the theory of orbifold curves from the perspective of stacks. By an orbifold curve I mean something like a $1$-dimensional irreducible Deligne-...
Alekos Robotis's user avatar
4 votes
0 answers
140 views

Canonical comparison between $\infty$ and ordinary derived categories

This question is a follow-up to a previous question I asked. If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
Stahl's user avatar
  • 1,089
2 votes
0 answers
125 views

Classifying stack for finite flat group scheme

Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...
mhahthhh's user avatar
  • 381
2 votes
1 answer
216 views

Is a finite morphism of Deligne-Mumford stacks proper?

The situation that I am in is the following. Let $\mathcal{X}$ be a smooth Deligne-Mumford stack over a field $k$. Let $X$ be a $k$-scheme together with a morphism $\pi;\mathcal{X}\rightarrow X$ (you ...
Hajime_Saito's user avatar
2 votes
0 answers
113 views

What are the categories of IND and PRO schemes?

below is a mathexchange question with no answers so I drop it here. I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are, in particualer the relations with ...
Marsault Chabat's user avatar
5 votes
0 answers
142 views

Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks

I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...
Hajime_Saito's user avatar
3 votes
0 answers
132 views

Relationship between $\infty$-categories of ind-coherent sheaves on the base and total space

My question is vaguely as follows, let $G\to E\to X$ be a principal $\infty$-bundle for some group object $G$ in a category of ($\infty$-)stacks. Can we recover the category $\operatorname{IndCoh}(E)$ ...
Grisha Taroyan's user avatar
2 votes
0 answers
228 views

Finite generation of stack cohomology

Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra? For instance, $\text{H}^*(\text{B}\mathbf{G}...
Pulcinella's user avatar
  • 5,506
2 votes
0 answers
156 views

Dualizing sheaf for classifying stack and duality

For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...
E. KOW's user avatar
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2 votes
1 answer
283 views

Hypercover and hyper descent

I am trying to understand the descent condition using hypercovers. The condition says that a hyper cover of a scheme $X$ is a simplicial set $Y_{\bullet}$ that satisfies the condition $Y_n\rightarrow ...
Hello's user avatar
  • 23
1 vote
0 answers
167 views

Moduli stack of l-adic sheaves?

Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors. Let $\ell$ be a prime ...
user577413's user avatar
9 votes
0 answers
240 views

Grothendieck purity for Brauer groups of stacks

Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
Tim Santens's user avatar
1 vote
0 answers
187 views

An algebraic stack is an algebraic space if and only if it has the trivial stabilizer group

Let $G\to S$ be a smooth affine group scheme over a scheme. Let $U$ be a scheme over $S$ with an action of $G$. Let $[U/G]$ be the quotient stack. In Alper's note: Stacks and Moduli, there is a result ...
Yuen's user avatar
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1 vote
1 answer
193 views

Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface

Let $k$ be an algebraically closed field of $\text{ch}(k) =0$. Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack. Let $\mathbf{P}(1,1,2)$ be the weighted ...
Z.N's user avatar
  • 11
4 votes
0 answers
166 views

Confusion in identification of quasicoherent sheaves on BG and G -representations

I asked this question on MSE a few days ago, but didn't get a response and also managed to confuse a senior colleague with it since then. This is probably a stupid question, so please bear with me. ...
Sergey Guminov's user avatar
1 vote
0 answers
80 views

Algebraizable image of a morphism of Galois cohomology stacks

Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...
kindasorta's user avatar
  • 1,631
2 votes
0 answers
271 views

Grothendieck duality for root stacks

Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be a projective scheme over $k$, let $D = \sum_{i=1}^d$ be a simple normal crossing divisor. Let ${\bf a} = (a_1,\cdots,a_d)$ be ...
Walterfield's user avatar
0 votes
1 answer
248 views

On sheaf quotient

Let $X$ be a scheme and $G$ be a group acting on $X$. Suppose the action is not free. Consider the quotient sheaf $X/G.$ Can we directly prove that the sheaf quotient is not an algebraic space?
S.D.'s user avatar
  • 492
2 votes
0 answers
168 views

$D^b_\text{Coh}(X)$ for a smooth, proper Deligne-Mumford stack

Let $X$ be a smooth, proper DM stack over a field $k$. I see in Hall-Rydyh's paper "Perfect Complexes on Algebraic Stacks" (https://arxiv.org/abs/1405.1887) a discussion of compact ...
locally trivial's user avatar
1 vote
0 answers
140 views

On the stack of bundles

Let $K$ denote the function field $\mathbb C((t))$ and let $X$ be a smooth projective curve of genus $g\geq 2$ over $K$. Let $r\geq 2$ be some positive integer. Let $B$ denote the moduli of vector ...
S.D.'s user avatar
  • 492
1 vote
0 answers
190 views

How to define Cartier divisor and Weil divisor on algebraic stack?

How to define Cartier divisor and Weil divisor on algebraic stack? Do they correspond to line bundles on stack like the case of schemes? In case of a Deligne-Mumford stack, can we have a simpler ...
user124771's user avatar
0 votes
1 answer
97 views

Covering a stack by an open substack that contains all points of finite type

Let $X = \text{Spec}A$ be an affine scheme. It is well known that if $U$ is an open subset which contains $\text{SpecMax}A$, then $U\supseteq X$. The previous statement generalizes to arbitrary ...
kindasorta's user avatar
  • 1,631
1 vote
0 answers
75 views

Function vanishing on the image of a morphism of algebraic stacks

Let $f: X\longrightarrow Y$ be a morphism of algebraic stacks, and assume I know the Krull dimension of the ring of global functions on $Y$ to be $n$. Assume that the stack $X$ has "stacky" ...
kindasorta's user avatar
  • 1,631
1 vote
0 answers
175 views

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 ...
mhahthhh's user avatar
  • 381
0 votes
0 answers
134 views

Cartesian square in the category of Algebraic stacks

Suppose we have a commutative diagram of Artin stacks $ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\...
S.D.'s user avatar
  • 492
0 votes
0 answers
147 views

Topological property of an algebraic stack and its presentation

I started to learn algebraic stacks this January. I found there are several properties of algebraic stacks which are defined in terms of their underlying topological spaces, for example, connectedness,...
mhahthhh's user avatar
  • 381
1 vote
0 answers
149 views

Moduli of morphisms between varieties

Let $\mathcal{M}, \mathcal{N}$ be two "well-behaved" (i.e. representable by an algebraic stack parametrising flat, proper, surjective, finitely presented morphisms with geometric fibres ...
Matthias's user avatar
  • 203
2 votes
0 answers
161 views

Closed embedding into weighted projective stack/space

Let $ Z $ be a proper or even, projective scheme. I have two related questions. (everything is over complex numbers) (1) What is a criteria for a map $ \phi: Z \rightarrow \left[ \mathbb{A}^{n+1} - (0)...
Cranium Clamp's user avatar
3 votes
0 answers
106 views

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 (...
Pulcinella's user avatar
  • 5,506
8 votes
2 answers
290 views

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 ...
RandomMathUser's user avatar
6 votes
0 answers
207 views

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 (...
Daniel Loughran's user avatar
2 votes
0 answers
161 views

Explicit computation of inertia stacks

I am learning algebraic stacks myself with some reference recommended by my friends. I know from here Section 6.1 that an explicit description of the fibre product of stacks $\mathfrak{X}\times_\...
mhahthhh's user avatar
  • 381
5 votes
1 answer
310 views

How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point

I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof. Setting Let $\mathcal{M}_{1, 1, k}$ ...
PIELEO13's user avatar
8 votes
2 answers
290 views

Smallest atlas for algebraic stack

Let $X$ be an algebraic stack of finite type over a field. Is there an intrinsic way to calculate the minimum of the dimensions of all atlases of $X$? By intrinsic here I mean using constructions such ...
NZK's user avatar
  • 345
5 votes
0 answers
158 views

Is $\operatorname{Rep}(G,\operatorname{SL}_2)$ representable by an algebraic space?

Let $G$ be a finite group. Consider the category of rigid analytic spaces over $\operatorname{Spf}\mathbb{Q}_p$, and let $\operatorname{Rep}(G, \operatorname{SL}_2)$ be the fibred category above it, ...
kindasorta's user avatar
  • 1,631
2 votes
0 answers
68 views

a connected geometrically unibranch algebraic stack of finite type over a field is irreducible

Let $f:X\to \mathfrak{X}$ be a smooth presentation of geometrically unibranch connected algebraic stack by a scheme, which is geometrically unibranch since being geom. unibranch is local in smooth ...
mhahthhh's user avatar
  • 381
1 vote
0 answers
146 views

Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?

I need the reference to a detailed proof the following fact. Let $g\geq 2$ be an integer. Let $\mathcal M^{ss}_g: Sch\rightarrow Groupoids$ be the prestack which sends a scheme $T$ to the groupoid of ...
S.D.'s user avatar
  • 492
1 vote
0 answers
174 views

Quotient stack is an algebraic space when $G$ is finite and acts freely

I have been following Jarod Alper's lecture series on YouTube on Stacks https://youtube.com/playlist?list=PLhFI5R_xInjdhtWuhgYlA8NZGXO-unnl4 From what I understand - If a smooth affine group scheme $...
angry_math_person's user avatar
1 vote
0 answers
300 views

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 ...
Yuhang Chen's user avatar
  • 1,099
2 votes
1 answer
292 views

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, ...
Yuhang Chen's user avatar
  • 1,099
1 vote
1 answer
278 views

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 ...
Yuhang Chen's user avatar
  • 1,099
1 vote
0 answers
188 views

When is a quotient stack of finite type?

Let $k$ be a field. Let $X$ be a scheme over $k.$ Let $G$ be an affine smooth group scheme over $k$ acting on $X.$ Suppose $X$ is of finite type over $k.$ Does this guarantee that the quotient stack $[...
Yuhang Chen's user avatar
  • 1,099
1 vote
1 answer
279 views

Birational morphisms from DM stacks to their coarse moduli spaces

Let $X$ be an integral scheme over a field. Let $G$ be a finite group acting on $X$ faithfully. Assume the quotient stack $[X/G]$ is separated (e.g., when $G$ acts on $X$ properly). Then $[X/G]$ is a ...
Yuhang Chen's user avatar
  • 1,099
4 votes
1 answer
574 views

Questions about root stacks

Let $\cal X$ be a DM stack and ${\cal D}\hookrightarrow{\cal X}$ an effective Cartier divisor on it. Suppose that $n$ is a positive integer invertible in ${\cal X}$. Let $\sqrt[n]{{\cal D}}\to{\cal X}$...
Carletto's user avatar
  • 163
5 votes
0 answers
246 views

Cohomology of coherent sheaves on Deligne Mumford stacks

Suppose that $\cal X$ is tame Deligne Mumford stack with generic trivial inertia. Let $X$ be its muduli space and $f:{\cal X}\to X$ the projection. Let $\cal F$ be a coherent sheaf on $\cal X$. Is it ...
Carletto's user avatar
  • 163
11 votes
0 answers
362 views

Moduli stacks of algebraic surfaces—obstructions to existence?

The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...
Pulcinella's user avatar
  • 5,506
1 vote
0 answers
125 views

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 \...
S.D.'s user avatar
  • 492
8 votes
0 answers
296 views

$\mathbb G_{\mathrm{m}}$-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 ...
W. Rether's user avatar
  • 405
1 vote
0 answers
471 views

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....
EJAS's user avatar
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