The stacks tag has no usage guidance.

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### Does Hom(X,Spec R) = Hom(R, O(X)) hold for algebraic stacks?

For an affine scheme $Spec R$ and a scheme $X$ we know that $Hom(X,Spec R) = Hom(R,\Gamma(X,\mathcal{O}_X))$.
Does it still hold when we replace $X$ with an algebraic stack?
My guess is yes as ...

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**1**answer

1k views

### surjective morphism of schemes or epimorphism of sheaves?

I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...

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213 views

### properness of stack

Hi,
assume we have an algebraic stack $A$ over $Sch(\mathbb{Z})$ which is quasi-compact and with separated diagonal. Assume that I have a stack $B$ which is obtained by rigidifying $A$ by a subgroup ...

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901 views

### Are non-algebraic stacks useful in algebraic geometry?

The title is a bit vague. What I want to know is if there is any geometric application of non-algebraic stacks. I know e.g. the category of coherent sheaves is an example. But I want to ask if people ...

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218 views

### Derived pullback of quasi-coherent complexes between algebraic stacks

Let $X$ and $Y$ be Artin algebraic stacks, and $f:X\to Y$ a morphism. I am interested in a pullback morphism
$Lf^\ast : D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$.
In his paper Sheaves on Artin stacks, ...

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**1**answer

297 views

### can one define the pullback between stacks of coherent sheaves for non-flat morphisms?

Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}_X, \mathcal{M}_Y$.
Does one have for free an induced pullback ...

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401 views

### Algebraic stacks as (etale) groupoid alg.spaces/schemes

Assume given an algebraic stack() $\mathcal{X}$ with presentation $X_0 \to \mathcal{X}$, and the corresponding groupoid $X = (X_0\times_\mathcal{X} X_0 \rightrightarrows X_0)$ in algebraic spaces (or ...

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**1**answer

459 views

### Query on comment in Deligne-Mumford (1969)

In Deligne and Mumford's famous 1969 paper, The irreducibility of the space of curves of given genus, definition 4.6 (that of algebraic stacks) has the following footnote:
This definition is the ...

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**2**answers

549 views

### Reference for moduli stack of principal G-bundles?

Hi,
I'm looking for a reference for the fact that the moduli stack $M_{GL_r,X}$ of $GL_r$-bundles over $X$ is an algebraic (Artin) stack. I'm only interested in the case where $X$ is a curve (for ...

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397 views

### Stack cohomology vs cohomology of the quotient

Let $X$ be an affine scheme over an algebraically closed field $K$ of positive characteristic, $G$ a finite group acting on $X$, $[X/G]$ the quotient stack and
$p:[X/G]\to X/G$ the natural map of ...

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423 views

### (Sh,Sh-map) represents the category of sheaves on a stack.

I'm trying to understand the following theorem, but I don't think I'm reading it correctly.
Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid of ...

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**3**answers

2k views

### Is every algebraic space the quotient of a scheme by a finite group?

In this MO question it is claimed that a catchphrase for "algebraic spaces" could be that they are "the result of looking at the orbit space of the action of a finite group on a scheme".
Hence my ...

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**2**answers

2k views

### The different types of stacks

This question is very naive, but it will help me a lot in getting in to the vast literature about stacks.
The question is this: there are many kinds of stacks (algebraic spaces, DM, algebraic stacks, ...

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**4**answers

725 views

### Adjunctions form a stack

Let $C$ be a base category, $F,G$ be two categories fibered over $C$ and $F \to G$ be a morphism. The following criterion is used very often: If all the fiber functors $F_U \to G_U$ ($U \in C$) are ...

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**1**answer

680 views

### Basic questions about stacks 2

I have again three basic questions about stacks.
1) When we consider categories fibered in groupoids, do we always mean small or essentially small groupoids? Especially I want to know if algebraic ...

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**1**answer

315 views

### Irreducibility of quotient stacks.

Let $[X/G]$ be a quotient stack such that $X$ is irreducible and $G$ acts trivially on $X$ (I am just adding automorphisms to every point). Under which hypothesis is $[X/G]$ irreducible as an Artin ...

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**1**answer

1k views

### Qcoh(-) algebraic stack?

The $2$-functor $\text{Qcoh} : \text{Sch}^{op} \to \text{Cat}$, which sends a scheme to its category of quasi-coherent modules, is a stack by Descent Theory. Is it actually an algebraic stack? If not, ...

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**1**answer

623 views

### “Approximating” $BGL(1)$ by projective spaces

Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant ...

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469 views

### Automorphism groups and etale topological stacks

Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a ...

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525 views

### Coarse Moduli space of proper Deligne-Mumford stacks

Let F be a finite type proper Deligne-Mumford Stack over a perfect field. Is it true that the coarse moduli space of F is proper?

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**1**answer

658 views

### What is the stalk of a stack?

When we study sheaves of sets (on a space X or a site C) we are often interested in the stalks of the sheaf (at either a point $p:1\to X$ or a left exact, cover-preserving functor $a:C\to Sets$). I ...

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446 views

### Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...

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**1**answer

384 views

### Ramification formula for orbifolds

It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this ...

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829 views

### Basic questions about stacks

I'm trying to understand some basics of stacks in algebraic geometry and have three questions:
1) As far as I understand, the moduli stack of vector bundles over a scheme $X$ is a replacement for the ...

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**1**answer

533 views

### Does a degeneration always have a larger-dimensional automorphism group?

Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an isotrivial degeneration of $\xi$). ...

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749 views

### Exactly how is 'the diagonal is representable' used for algebraic stacks…

...apart from stating properties of $(s,t):X_1 \to X_0\times X_0$ for the a presenting algebraic groupoid $X_1 \rightrightarrows X_0$?
Once we know that given a stack $\mathcal{X}$ we have a smooth ...

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**1**answer

397 views

### Categories of descent data

Let us work over the etale site $\mbox{Aff}/S$ (for the sake of definiteness) for some fixed base scheme $S$, where the covers are jointly surjective etale maps $\{ U_i \rightarrow U\}_{i\in I}$ (and ...

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830 views

### What about stacks of categories in algebraic geometry? II

I've made this a new question, rather than expanding the first one.
Torsten gives a good answer, and it partially illustrates in practice the 'second approach' I outlined in my other question. (You ...

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497 views

### Local structure of Deligne-Mumford stacks

Let $\mathcal{X}$ be a separated Deligne-Mumford stack over an algebraically closed field $k$ and let $X$ be the corresponding coarse moduli space, which we assume to exist. There is a map ...

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### What about stacks of categories in algebraic geometry?

Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...

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383 views

### Is there a non-quotient stack with affine stabilizers whose good moduli space is a geometric point?

Definitions: One says that a map $\pi\colon\mathcal X\to X$ from an algebraic stack to an algebraic space is a good moduli space if $\pi$ is cohomologically affine and universal for maps to schemes. ...

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132 views

### Maximal algebraic sub-groupoids

By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in the latter case are ...

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417 views

### References for constructible sheaves on complex analytic stacks

I'm looking for references on constructible sheaves and the six operation formalism on analytic stacks (stacks fibered over complex analytic spaces). Does anyone have some suggestions?
Basically I ...

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865 views

### Why is lack of functoriality of the Lisse-Etale topology specific to the Lisse-Etale topology?

I'm trying to follow the explanation given in Olsson's "Sheaves on Artin stacks" for the lack of functoriality for lisse-étale topology: Let $f:Y \to X$ be a morphism of algebraic stacks. The functor ...

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149 views

### $G_m$-cohomology of a motif (that corresponds to a stack?)

As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth ...

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495 views

### Is there a “geometric” language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions.
Here, ...

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### Sheaves on stacks and interesting functors

Let $G$ be a finite group and $H \subset G$ a normal subgroup. Consider $G$, $H$, and $X=G/H$ as affine algebraic groups over some algebraically closed base field $k$.
I hear that there is an ...

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**1**answer

274 views

### Rational points of an algebraic space over finite field

If $X$ is an algebraic space of finite type over a finite field $k$, then I think it is true that the set of $k$ rational points of $X$ is finite.
This is of course true for $X$ is a scheme. I wish ...

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693 views

### Proving that a map is a morphism

Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map
we associate its branch divisor, ...

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517 views

### Two questions on algebraic stacks

Question 1: The main reference on algebraic stacks (Laumon and Moret--Bailly) defines a separable algebraic stack as one having universally closed diagonal. For schemes separability is simply defined ...

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297 views

### The plus construction for stacks of n-types

In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+1\right)$ times, and in general, ...

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292 views

### What is the signficance of the existence of a moduli stack to a moduli problem?

This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.
Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...

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211 views

### closed substacks of cartesian powers of a stack

Let $\mathbb{Z}/2\mathbb{Z}$ act on $\mathbb{A}^1$ as $x \mapsto -x$, and let $\mathscr{X}$ be the quotient stack. It has coarse moduli space $\mathbb{A}^1$ and a residual gerbe ...

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180 views

### When do maps of ineffective orbifolds descend to their effective part?

If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...

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### When is a category of groupoid schemes fibred over schemes?

The category of topological categories $Cat(Top)$ is fibred over $Top$ - the functor sending a groupoid $X_1 \rightrightarrows X_0$ to its object space $X_0$ is a Grothendieck fibration. Now one can ...

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241 views

### What condition on a “bibundle between categories” generalizes “right-principal bibundle between groupoids”?

My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at the ...

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923 views

### Applications of Stacks

I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) scheme-theoretic argument or theorem ...

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313 views

### Weak colimits of weak and strict presheaves in groupoids

Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and ...

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803 views

### Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?

If $X$ is an algebraic stack fibered in sets (and therefore essentially a sheaf), is it an algebraic space? It seems conceivable that at least when $X$ is Deligne-Mumford, it is actually an algebraic ...

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352 views

### Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory?

If I have a Deligne-Mumford stack $\Pi : X \to (\mathrm{Sch}/k)$ for some field $k$, can it be reconstructed from $\Pi^{-1}(C) \subset X$ for some "small" subcategory $C \subset (\mathrm{Sch}/k)$? For ...