# Questions tagged [algebraic-stacks]

The algebraic-stacks tag has no usage guidance.

283
questions

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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, ...

3
votes

0
answers

71
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### 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-...

4
votes

0
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140
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### 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\...

2
votes

0
answers

125
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### 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 $\...

2
votes

1
answer

216
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### 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 ...

2
votes

0
answers

113
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### 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 ...

5
votes

0
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142
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### 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 $\mathscr{X}$ ...

3
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132
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### 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 $\operatorname{IndCoh}(E)$ ...

2
votes

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228
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### 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}...

2
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156
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### 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 ...

2
votes

1
answer

283
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### 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 ...

1
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0
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167
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### 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 ...

9
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240
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### 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 (...

1
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0
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187
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### 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 ...

1
vote

1
answer

193
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### 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 ...

4
votes

0
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166
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### 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.
...

1
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0
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80
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### 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 ...

2
votes

0
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271
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### 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 ...

0
votes

1
answer

248
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### 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?

2
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### $D^b_\text{Coh}(X)$ 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 ...

1
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0
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140
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### 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 ...

1
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0
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190
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### 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 ...

0
votes

1
answer

97
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### 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 ...

1
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0
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75
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### 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" ...

1
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175
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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 ...

0
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134
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### 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}\ \ }\...

0
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147
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### 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,...

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149
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### 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 ...

2
votes

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161
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### 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)...

3
votes

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answers

106
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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 (...

8
votes

2
answers

290
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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 ...

6
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answers

207
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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 (...

2
votes

0
answers

161
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### 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_\...

5
votes

1
answer

310
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### 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 $\mathcal{M}_{1, 1, k}$ ...

8
votes

2
answers

290
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### 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 ...

5
votes

0
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158
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### 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, ...

2
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0
answers

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### 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 ...

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146
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### 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 ...

1
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0
answers

174
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### 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 $...

1
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0
answers

300
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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 ...

2
votes

1
answer

292
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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

1
answer

278
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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
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0
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188
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### 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 $[...

1
vote

1
answer

279
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### 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 ...

4
votes

1
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574
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### 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 $\sqrt[n]{{\cal D}}\to{\cal X}$...

5
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246
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### 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 ...

11
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0
answers

362
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### 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 ...

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125
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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 \...

8
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296
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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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471
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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....