Questions tagged [algebraic-stacks]

Use this tag for questions related to algebraic stacks, which are a stacks in groupoids X over the etale site such that the diagonal map of X is representable, and there exists a smooth surjection from a scheme to X.

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Show something is quasi-isomorphism by showing pullback is quasi-isomorphism.

My question is the following: Let $T$ be a topos and $X\to e$ be a covering of the final object $e\in T$ with respect to the canonical topology. We have a morphism of topoi $f\colon T/X\to T$ where $f^...
Enki's user avatar
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55 views

What are the categories of IND and PRO schemes?

I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are... Are they full subcategory of algebraic stacks? Are they full subcategory of algebraic spaces? Edit: I ...
Marsault Chabat's user avatar
1 vote
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49 views

Can one calculate pullback of prestacks fiberwise?

The pullbacks of presheaves can be computed pointwise, and such computation determines the pullback, i.e. $F = G\times_S H$ iff $F(X) = G(X) \times_{S(X)} H(X)$ for all X. Does similar reduction holds ...
Alexander Golys's user avatar
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1 answer
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Simplicial manifolds which do not satisfy Kan condition locally?

Similar to Kan condition for simplicial sets, there are also Kan condition for simplicial manifolds, that is, we ask the horn projection $p^k_j: X_k \to Hom(\Lambda[k,j], X)$ to be a surjective ...
Chenchang Zhu's user avatar
4 votes
1 answer
121 views

An algebraic stack as a non-linear analog of a complex of vector spaces

In their paper Derived Quot Schemes Kapranov and Ciocane-Fontanine write in the introduction: Indeed, an algebraic stack is a nonlinear analog of a complex of vector spaces situated in degrees $[-1,0]...
Margaret's user avatar
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2 answers
94 views

Doubt about the definition of fine moduli space

I’m studying “Introduction to differentiable stacks” (Grégory Ginot) and I don’t understand a technicality in the (somewhat informal) definition of fine moduli space given at page 12. Basically if we ...
Kandinskij's user avatar
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2 votes
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45 views

Describe morphisms of quotient stacks.

Let $G,H$ be group schemes, acting on schemes $X,Y$ via $G\times X\to X$ and $H\times Y\to Y$. I want a description of a morphism $[X/G]\to [Y/H]$. When $X$ is a point, we have $[X/G]=\mathrm BG$, the ...
Display Name's user avatar
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Smoothness of Artin Stack by Dimension Estimation

For a proper scheme or, more general, a Deligne-Mumford stack $X$ over an algebraically closed field $k$ it is known that if at all points $x$ the dimension $dim_x X$ of the local ring at that point ...
Matthias's user avatar
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4 votes
1 answer
92 views

Sheaf cohomology on quotient stacks

Suppose I have a scheme $X$ over $\mathbb C$, acted on by a finite group $G$. Let $\mathcal F$ be a $G$-equivariant coherent sheaf of $\mathcal O_X$-modules. Then I can form the stack quotient $[X/G]$,...
stacklearner's user avatar
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25 views

Clarifying an example for quotient stacks: Whether diagonal is closed substack

In trying to remember an example about quotient stacks, I think I've got something turned around. I am trying to determine whether the diagonal is a closed substack of the (product of) quotient stack(...
locally trivial's user avatar
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Confusion in identification of quasicoherent sheaves on $BG$ and $G$-representations

Let $k$ be some field, say of characteristic 0, and let $G$ be a finite group considered as a discrete algebraic group. Then we get the classifying Deligne-Mumford stack $BG$ and its etale ...
Sergey Guminov's user avatar
2 votes
0 answers
28 views

Representable morphisms of algebraic spaces

Let $X,Y$ be algebraic spaces, and let $X\longrightarrow Y$ be a representable morphism. Let $S$ be a scheme and let there be a morphism $S\longrightarrow Y$. The fibre product, $X\times_Y S$, is said ...
kindasorta's user avatar
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2 votes
1 answer
179 views

The gap between algebraic spaces and DM-stacks

I am following Jarod Alper's course "Introduction to Stacks and Moduli". He gives the following definitions: An algebraic space is a sheaf $X$ on $\mathrm{Sch}_{Et}$ such that there is a ...
Sergey Guminov's user avatar
3 votes
1 answer
100 views

An etale morphism to a separated scheme is affine?

I see the following statement in the proof of the local structure theorem of DM stacks from Jarod Alper's notes (Theorem 4.2.1). My question is why an etale from an affine scheme to a separated ...
user393795's user avatar
2 votes
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References on blowing up algebraic stacks.

The title says the question. I want to know if there are any texts about blowing up for stacks, including definition and necessary purity checks.
Display Name's user avatar
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Stackification of finite categories

Assume I have a base category $\mathscr S$ with finite limits and a geometric morphism $\gamma :\mathscr S\to Fin$ into the category of finite sets (for example because $\mathscr S$ is positive ...
Nico's user avatar
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6 votes
2 answers
219 views

Definition of stabilizer of a point in a stack

I was reading in Adeel Khan's ''A Modern Introduction to Algebraic Stacks'' and came across the definition of the stabilizer of an $R$-valued point $x$ of a stack $X$. My question will be about that ...
Daniël Apol's user avatar
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1 vote
1 answer
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Categories fibered in groupoids and the slice category

I read that one example of categories fibered in groupoids is the slice category $\mathcal{C}_{/x}\to \mathcal{C}$ where $x\in \mathcal{C}$ an object. But as the usual definition of slice category ...
Chanel Rose's user avatar
1 vote
0 answers
92 views

Flatness & surjectivity for Group Scheme Morphism

I am currently reading https://arxiv.org/abs/math/0703310 and I was wondering why the map $S \to B_SG''$ in proof of proposition 2.7 (c) is faithfully flat. This is as far as I already understood ...
max_121's user avatar
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Confusion regarding valuative criteria for proper maps

Let $f : X \to Y$ be a proper morphism of schemes i.e. universally closed, finite type, separated. Then the valuative criteria states that it is iff the condition that given a commutative diagram of ...
Angry_Math_Person's user avatar
1 vote
0 answers
37 views

representing compactified stack as a global quotient

Let $M$ be a DM stack over a field $k$. Assume I managed to represent it as $[X/G]$ for some scheme $X$ and a group scheme $G$. Does that imply that the DM compactification $\overline{M}$ of $M$ is ...
iou's user avatar
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2 votes
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41 views

Points of a quotient stack as a section of a contracted product

Let $G$ be an algebraic group acting on an $S$-scheme $X$. Let $T\in \operatorname{Sch}/S $. Given a principle $G$-bundle $P\to T$, the contracted product is defined as $(P\times_SX)/G$ where $G$ acts ...
Conjecture's user avatar
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Fpqc sheaf property of algebraic stacks

It is known that algebraic spaces are fpqc sheaves. A proof by Gabber can be found in the Stacks Project at tag 0APL. Naturally one may ask whether this generalizes to algebraic stacks. Given an ...
SmileLee's user avatar
4 votes
0 answers
67 views

What is the inverse limit of algebraic stacks?

Let $S$ be a base scheme. Let $(F_{i},f_{ii'})_{i\in I}$ be a directed inverse system of algebraic spaces over $S$. Then if each $f_{ii'}$ is affine, the inverse limit $\lim_{i}F_{i}$ exists as an ...
Toney Leung's user avatar
4 votes
2 answers
158 views

Naive Line Bundle Functor isn't a Zariski Sheaf

I'm reading a set of notes on algebraic stacks by Anatoly Preygel, and in the opening paragraphs it says Meanwhile moduli problems... often have the property that the functor $$T \mapsto \{ \text{...
Chris Grossack's user avatar
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138 views

When is a stack a sheaf?

On big etale site we have the slogan "Schemes are sheaves, sheaves are stacks" Now we know there are sheaves which are not schemes, in fact a sheaf is a scheme iff it is representable as a ...
Angry_Math_Person's user avatar
1 vote
1 answer
139 views

Action of the stabilizer group on the completed local ring

Let $\mathcal{M}$ be a DM stack and $M$ its coarse moduli space. Then we have the map $\mathcal{M} \to M$. Choose a geometric point $x \in M$ and a geometric point $\bar{x} \in \mathcal{M}$ which lies ...
iou's user avatar
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Differences between equivalent definitions of algebraic spaces

I have trouble understanding the differences between the following two equivalent definitions of algebraic spaces. Let $k$ be a field. The first one is An algebraic space is an '{e}tale sheaf $(Sch/S)...
Toney Leung's user avatar
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154 views

Understanding the quotient (pre)stack

Let $G\to S$ be a group scheme acting on a scheme $X\to S$. Define a prestack $[X/G]$ whose objects over an $S$-scheme $T$ are pairs $(P\to T, P\xrightarrow{\varphi} X)$ where $P\to T$ is a $G$-torsor ...
Nico's user avatar
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3 votes
1 answer
99 views

How to compute the fiber of point between algebraic stacks?

Suppose $f: \mathcal{X}\rightarrow Y$ is a morphism from an algebraic stack $\mathcal{X}$ to a scheme $Y$. It can induce a unique topological morphism $|f|:\mathcal{|X|}\rightarrow Y$. Suppose $y\in Y$...
Vector's user avatar
  • 275
2 votes
1 answer
138 views

Quasi-separatedness of diagonal of DM stacks and group stabilizer of a point

I was studying Jarod Alper's book on stacks (https://sites.math.washington.edu/~jarod/moduli.pdf) and after the definition of stacks and some basic discussion about them, he proposes the following ...
Luca Morstabilini's user avatar
1 vote
0 answers
60 views

Understanding the representations of stabilizer groups of points in a stack

I'm trying to learn about stacks, but I'm sure I'm misunderstanding something about sheaves on them. Let $\mathcal X$ be a DM stack over $\mathbb C$, and let $F$ be a sheaf of $\mathcal{O}_{\mathcal X}...
stacklearner's user avatar
1 vote
0 answers
173 views

How to compute the fiber product of stacks?

Update: I'm now at this step: I want to confirm that if $b:P\to T$, then $[U/G](b)((P,f))$ is $P\times_TP$ and $P\times_TP\simeq P\times G$ Specifically, I want to understand in where $\mathcal{P}$ ...
Hypatia du Bois-Marie's user avatar
4 votes
1 answer
119 views

What's the explicit description of the atlas of quotient stack?

Let $U\to[U/G]$ be a quotient stack. How does one associate a $G$-torsor $P\to S$ and a equivariant map $P\to U$ to each $S\to U$?
Hypatia du Bois-Marie's user avatar
0 votes
1 answer
172 views

Compactification of $M_{1, 1}$

Let $M_{1, 1, \mathbb{Z}}$ be a moduli stack of elliptic curves. Denote $M_{1, 1, k}$ its base change to a field $k$. We have the map $j: M_{1, 1, k} \to \mathbb{A}^1_{k}$ which realizes the affine ...
iou's user avatar
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1 vote
0 answers
37 views

Pullback of a line bundle on a stack amounts to a rational character

Let $m,q$ be positive integers. Let $\mathcal M$ be a stack and $\mathcal L\in \operatorname{Pic}(\mathcal M)\otimes \Bbb Q$. Let $B\Bbb G_m^q:=[\operatorname{Spec}\Bbb Z/\Bbb G_m^q]$ over a field $k$ ...
Conjecture's user avatar
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0 votes
0 answers
122 views

Reference request: prestacks

Prestacks do not arise often in nature so people do not write about their properties, sadly I found one in the wild. I've tried finding a reference which proves basic properties of prestacks, but all ...
Mathmop's user avatar
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1 vote
0 answers
72 views

Is the moduli stack of stable maps functorial?

Let $f:X\rightarrow Y$ be a morphism where $X$ and $Y$ are smooth projective varieties over $\mathbb{C}$. Does $f$ induce a well-defined "pushforward" morphism $\overline{\mathcal{M}}_{g,n}(...
Chi Hong Chow's user avatar
2 votes
1 answer
99 views

Residual Gerbe is Étale Locally a Classifying Stack

I heard from someone that, if $x:\text{Spec}k\to\mathcal{X}$ is a point of an Deligne-Mumford stack (algebraic should be OK, I am only assuming Deligne-Mumford so we know the residual gerbe exists), ...
user avatar
2 votes
0 answers
145 views

Is the pullback of a quotient stack a quotient stack?

Let's say I have a $U$-scheme $X$ and a group $U$-scheme $G$ acting on $X$. Then I can consider the quotient stack $[X/G]$ on $\text{Sch}/U$. The $T$-objects of this stack are simply $G$-torsors $P\to ...
user avatar
3 votes
0 answers
103 views

Isomorphism Sheaves of an Algebraic Stack

I am struggling to find information on the Stacks project, so I am crowdsourcing here. (Sorry.) Let $\mathcal{X}$ be a stack (algebraic, if it makes a difference) over a category $\mathcal{C}$ (e.g. $\...
user avatar
0 votes
1 answer
87 views

In what sense stack (from category theory) is the (fibred) category as wikipedia asserts?

I am reading wikipedia article on the stacks https://en.wikipedia.org/wiki/Stack_(mathematics) and it contains assertion: The intuitive meaning of a stack is that it is a fibred category such that &...
TomR's user avatar
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2 votes
1 answer
355 views

Principal and Vector bundles over stacks

I am reading some papers concerning the moduli stack of vector bundles and there is some notion that I don't understand. Let us consider $\text{Vect}_{n}$ the stack of vector bundles with isomorphisms ...
Samantha Smith's user avatar
0 votes
0 answers
104 views

Universal Family in the stack of principal bundles

I have read that a stack can just be thought of as a solution to a moduli problem, and there are some examples, like the stack of elliptic curves or the stack of vector bundles over a scheme $X$ such ...
Samantha Smith's user avatar
2 votes
0 answers
321 views

Applications of the Chinese remainder Theorem to the study of the Hilbert scheme of points and $(\mathfrak{m},l)$-squeezed ideals.

The following construction gives a relation between the Chinese Remainder Theorem (CRT), the Noether nomalization lemma (NNL) and cofinite ideals in finitely generated $k$-algebras. Let $k$ be any ...
hm2020's user avatar
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0 votes
0 answers
144 views

Relation between quotient stacks and quotient sheaves

I am studying stacks and I am experiencing some troubles understanding quotient stacks. If I have a scheme $X$ where a group scheme acts $G$ (let's suppose that everything is defined over an ...
Samantha Smith's user avatar
2 votes
0 answers
221 views

Stacks and Schemes

I want to ask a very general question because I am in the middle of my bachelor thesis and I need some help. If we have a morphism from a scheme to a stack, what properties of the stack can be deduced ...
Samantha Smith's user avatar
3 votes
0 answers
122 views

Properness of the coarse moduli map in Keel-Mori theorem.

Given a stack $\mathscr X$ with enough assumptions we obtain a map $\rho: \mathscr X \to X$ to a coarse moduli space. Furthermore, $\rho$ is proper. I do not understand what it means for $\rho$ to be ...
trystero's user avatar
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2 votes
0 answers
68 views

How are sheaves/stacks viewed as generalised spaces?

Whenever I read literature that deals with higher categories, there is a point of view that sheaves/stacks are generalised spaces. What does that mean? For me, a space is a place to draw things. A CW ...
Isomorphism's user avatar
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1 vote
1 answer
255 views

Cech nerve and descent data

When generalizing from sheafs on a site to 2-sheafs or stacks, it is useful to first rephrase the descent data for ordinary (pre)sheafs in terms of the Cech nerve of a coverage (e.g. https://ncatlab....
NDewolf's user avatar
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