The stacks tag has no wiki summary.

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### Stacks in the fpqc topology

This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and ...

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### Does the concept of a basis for a topology on a category exist?

If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the ...

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### Universal property of X//G?

Given an operation of say a topological group on a topological space, one can form the quotient stack X//G:
the stack associated to the action groupoid.
Does this stack satisfy some kind of universal ...

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### When is a stack (NOT) geometric?

Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ ...

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### Double Category of Topological Stacks

There are two equivalent ways of describing topological stacks.
One is the "stacky" definition, that is, a topological stack is a stack $\mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if ...

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### Categorical construction of the category of schemes?

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...

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### Cotangent bundle of a differentiable stack

If you ever wanted to construct the tangent bundle of a differentiable stack, it's relatively simple:
First, if $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$, you could just say ...

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### Ordinary cohomology of stacks

Let $\mathbf{X}$ be a stack over $Top$ (a lax sheaf of groupoids, or some such thing). If it admits a surjective representable map $F \to \mathbf{X}$ then one can form the iterated fibre product to ...

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### Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to \left\[G\right]$, where $\left\[G\right]$ is the ...

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### finite etale covering of stacks

If $Y \to X$ is a finite etale map of schemes, then there exists a finite Galois morphism $Z \to X$ (i.e. it's a $Aut(Z/X)$-torsor) that factors as $Z \to Y \to X.$ The case when $X$ is normal is easy ...

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### Stacks in the Zariski topology?

I have two naive questions about stacks.
1) Is it possible to define stacks in the Zariski topology?
Presuming you can:
2) If a stack has a coarse moduli, and the coarse moduli space is a ...

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### What is the local structure of a Lie groupoid?

A manifold is locally $\mathbb R^n$. An orbifold is locally $\mathbb R^n/\{\text{finite group}\}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 \rightrightarrows ...

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### Why do gerbes live in H^2 ?

Line bundles on a scheme $X$ live in $H^1(X,O_X^*)$, where $O_{X}^{*}$ is the sheaf of invertible functions. If $X$ is noetherian separated, then we can think of this $H^1$ to be Cech cohmology ...

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### Conditions for “bootstrapping” a smooth DM stack?

In the preprint "Smooth toric DM stacks", Fantechi, Mann and Nironi define the stacks of their title, and show that each of these can be obtained through the following sequence of steps:
1) start ...

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### Positivity in stack geometry

I was wondering how much of the theory say of Lazarsfeld books can be carried to algebraic stacks (if this has been done).
Do we have a sensible notion of an ample (big, nef) line bundle? Of an ample ...

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### What can we do with a coarse moduli space that we can't do with a DM moduli stack?

A couple weeks ago I attended a talk about the Keel-Mori theorem regarding existence of coarse moduli spaces for Deligne-Mumford stacks with finite inertia. Here are some questions that I have been ...

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### Representation of Groupoids

The title is vague, my actuall question is the following:
Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? ...

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### In what topology DM stacks are stacks

Background/motivation
One of the main reason to introduce (algebraic) stacks is build "fine moduli spaces" for functors which, strictly speaking, are not representable. The yoga is more or less as ...

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### Degrees of etale covers of stacks

This is probably pretty basic, but as I said before I'm just beginning my way in the language of stacks.Say you have an etale cover X->Y of stacks (in the etale site). Is there a standard way to ...

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### Different interpretations of moduli stacks

I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to ...

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### fpqc covers of stacks

Artin has a theorem (10.1 in Laumon, Moret-Bailly) that if $X$ is a stack which has separated, quasi-compact, representable diagonal and an fppf cover by a scheme, then $X$ is algebraic. Is there a ...

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### how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?

So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...

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### Approximation of stacks / algebraic spaces

Let $B$ be a ring which is the colimit of rings $B_\lambda$. Let $X_\lambda$ be a stack (not necessarily algebraic) over $B_\lambda$ such that $X_\lambda \times_{B_\lambda} B_\mu = X_\mu$ and let $X ...

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### Stacks and sheaves

I'm a bit confused by the double role which sheaves play in the theory of stacks.
On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other ...

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### Is there a good notion of `Separated Stack'?

A scheme is separated if the diagonal inclusion $X \to X \times X$ is a closed immersion. I what to know if there is a good generalization of `separated' for algebraic stacks?
My usual stack ...

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### What is a proper stack?

I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism ...

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### coarse moduli space of DM stacks

This is related to another one of my questions on DM stacks. In Brian Conrad's article 'The Keel-Mori Theorem via Stacks', a sufficient condition on for an Artin stack to have coarse moduli space is ...

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### Is the inertia stack of a Deligne-Mumford stack always finite?

Let X be a DM stack over a field k. We follow the definition in Laumon and Moret-Bailly's book, so that its diagonal is quasi-compact (and hence diagonal is of finite type). Then is the diagonal ...

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### Algebraic versus Analytic Brauer Group

Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$. Then I think that someone (Serre?) showed that the Cohomological Etale Brauer Group agrees with the torsion part of the Analytic ...

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### D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...

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### Why are non-singleton covering families often ignored?

It seems to me that frequently when discussing stack conditions and descent, people consider only singleton covering families, i.e. there is some single covering map $U\to X$, for which one constructs ...

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### Sites which are stacks over themselves

A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a ...

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### Descend finite etale algebras

Let $\pi:X\to\mathcal X$ be a presentation of an Artin stack $\mathcal X$ of finite type over a field $k,$ and let $f:Y\to X$ be a finite \'etale covering. Does there exist a finite \'etale covering ...

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### Is the Torelli map an immersion?

The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description which works for ...

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### Near Trivial Quiver Varieties

So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup:
I've been looking at the simplest case that didn't look ...

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### Does sheafification preserve sheaves for a different topology?

Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 \cup T_2$ (by which I mean the smallest Grothendieck topology on $C$ containing $T_1$ and $T_2$). ...

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### Moduli stack of principally polarized abelian varieties

I'm looking for an accessible reference for the fact that the moduli stack of principally polarized abelian varieties is in fact an algebraic stack. Faltings/Chai sketch two possible proofs in their ...

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### When can cohomology be calculated on the coarse moduli space?

Suppose $\cal{X}$ is a DM-stack, and X its coarse moduli space. Let F be a sheaf on $\cal{X}$, and $\pi : \mathcal{X} \to X$ the projection. In all examples I have seen, it has been true that
...

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### Is there any Grothendieck Riemman Roch theorem for general stack ?

It seems that there is no g.r.r for stack yet according to dejong. Does anyone know anything about it? But as you might know, there are some complex manifold which is not scheme having atiyah singer ...

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### Local Joyal-simplicial presheaves?

It is well known that left Bousfield localizations of the global functor model category $Func(C^{op}, SSet_{standard})$ of functors with values in simplicial sets equipped with the standard model ...

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### k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?

If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and ...

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### Examples of algebraic stacks without coarse moduli space?

Keel-Mori's theorem says an algebraic stack with a finite diagonal over a scheme S has a coarse moduli space. What is an example of an algebraic stack without coarse moduli space?

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### What are Log Stacks

So, I've been running in both stacky circles and logarithmic circles and I've been wondering: is there a definition of log stack that is "useful"? I can imagine two such definitions:
1) A log stack ...

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### Good introductory references on algebraic stacks?

Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...

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### What are some examples of coarse moduli spaces?

It took me some effort to work out Gerashenko's nice simple example Can a singular Deligne-Mumford stack have a smooth coarse space? of a DM stack non-equisingular with its coarse moduli space, which ...

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### Does every morphism BG-->BH come from a homomorphism G-->H?

Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, ...

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### Kodaira-Spencer Theory and moduli of curves

I was looking at a paper of Farkas and the following confusing point came up.
Let $\mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $\pi: \mathscr{C} \to \mathscr{M}_g$ be the ...

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### Can a singular Deligne-Mumford stack have a smooth coarse space?

Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex ...

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### Are curves with `fractional points' uniquely determined by their residual gerbes?

One makes precise the vague notion of "curve with a fractional point removed" (see for instance these slides) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are ...

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### algebraic group G vs. algebraic stack BG

I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...