# 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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• 1,777
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 ...
• 3,719
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### 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 ...
• 1,373
45 views

### 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 ...
• 797
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 $[X/G]$,...
1 vote
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(...
1 vote
54 views

### 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 ...
• 4,038
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 ...
• 1,228
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 ...
• 4,038
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 ...
• 173
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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.
• 1,373
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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 ...
• 4,220
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 ...
• 4,122
1 vote
62 views

### 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 ...
• 181
1 vote
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 ...
• 759
1 vote
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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 ...
• 2,348
1 vote
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 ...
• 1,186
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 ...
• 3,118
95 views

### 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 ...
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 ...
• 739
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### 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{...
• 38.2k
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 ...
• 2,348
1 vote
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 ...
• 1,186
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1 vote
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### 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 $\mathcal{P}$ ...
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 $S\to U$?
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 ...
• 1,186
1 vote
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 $k$ ...
• 3,118
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 ...
• 468
1 vote
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• 701
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### 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 ...
• 5,693
1 vote