Questions tagged [algebraic-stacks]
The algebraic-stacks tag has no usage guidance.
280
questions
2
votes
1
answer
203
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 ...
2
votes
0
answers
108
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 ...
5
votes
0
answers
136
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 $\mathscr{X}$ ...
3
votes
0
answers
125
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 $\operatorname{IndCoh}(E)$ ...
2
votes
0
answers
227
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}...
2
votes
0
answers
152
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 ...
2
votes
1
answer
275
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 ...
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 ...
9
votes
0
answers
238
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 (...
6
votes
1
answer
661
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 ...
1
vote
0
answers
173
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 ...
4
votes
0
answers
158
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.
...
1
vote
1
answer
189
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 ...
2
votes
0
answers
500
views
Fourier-Mukai transforms on stacks
I have "generic" questions about Fourier-Mukai transforms.
Question 1: Does there exist a well-defined notion of Fourier-Mukai transform on (Deligne-Mumford) stacks?
Question 2: Do there exist ...
2
votes
0
answers
265
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 ...
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 ...
0
votes
1
answer
243
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?
2
votes
0
answers
167
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 ...
1
vote
0
answers
139
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 ...
1
vote
0
answers
183
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 ...
0
votes
1
answer
96
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 ...
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" ...
1
vote
0
answers
171
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 ...
8
votes
2
answers
284
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 ...
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}\ \ }\...
0
votes
0
answers
143
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,...
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 ...
2
votes
0
answers
155
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)...
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 (...
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 (...
2
votes
0
answers
154
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_\...
5
votes
1
answer
294
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 $\mathcal{M}_{1, 1, k}$ ...
8
votes
2
answers
289
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 ...
11
votes
0
answers
361
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 ...
19
votes
4
answers
1k
views
Are there prominent examples of operads in schemes?
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...
5
votes
0
answers
156
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, ...
2
votes
0
answers
67
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 ...
4
votes
1
answer
507
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 ...
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 ...
1
vote
0
answers
172
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 $...
1
vote
0
answers
298
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 ...
2
votes
1
answer
288
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, ...
1
vote
1
answer
277
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 ...
1
vote
0
answers
186
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 $[...
1
vote
1
answer
273
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 ...
4
votes
1
answer
544
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 $\sqrt[n]{{\cal D}}\to{\cal X}$...
5
votes
0
answers
232
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 ...
6
votes
1
answer
347
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 ...
8
votes
1
answer
663
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 ...
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 \...