The stacks tag has no wiki summary.

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### Quasi-coherent sheaves on classifying stacks

Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq ...

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### Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...

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### Cohomology of BG, algebraically

Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...

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### Several simple questions on the geometry of higher stacks

I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks.
For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...

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### Understanding the definition of the quotient stack $[X/G]$

I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles.
Explicitly, let $G$ be an affine smooth group ...

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### Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...

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### Algebraic stacks: limit preserving versus locally of finite presentation

I'm wondering what the precise relationship is between an algebraic stack being locally of finite presentation and being limit preserving. Under some mild hypotheses on the diagonal (in force ...

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### Groupoid as a 2-coequaliser

Let $G=(G_1, G_0, s, t, u, i,\circ)$ be a groupoid, where $s, t$ are source and target maps, $i$ is the inverse, $u$ is the unit, and $\circ$ is the composition.
Denote $\underline{G_1}, ...

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### Is “stackiness” transitive? (and a couple other basic questions about stacks)

Say, $B$ is a category fibered in groupoids over some category $C$, and $A$ is a category fibered in groupoids over $B$.
Suppose $A$ is a stack (over whatever site) over $B$, and $B$ is a stack over ...

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### The fibre product of two quotient stacks

My question is to know whether the fibre product of $[X/G]$ by $[Y/H]$ over a base scheme is $S$ is $[X \times_S Y/G \times H]$? And if yes, do you have any reference for it?
Thank you.

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### Stacks over diffeologies

Konrad Waldorf shows in his paper one may realize a Grothendieck topology on the category of diffeological spaces. Is there any work exploring stacks over the category of diffeologies?

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### $Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space

There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence ...

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### picard group of moduli of elliptic r-prym curves

Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves ...

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### fpqc stackification

I am reading Lurie's Tannakian paper (http://www.math.harvard.edu/~lurie/papers/Tannaka.pdf) and I am confused about one point.
At the end of page 3 he defines a stack-hom in any topos, which is ...

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### Differential Geometry of (Non-Abelian) Gerbes in the language of Brylinski

Context
In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there ...

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### Do Deligne-Mumford curves also have rational functions

If $X$ is a curve over a field of characteristic zero, then $X$ has a rational function, i.e., a finite morphism to the projective line.
Question. Suppose that $X$ is a Deligne-Mumford (or just ...

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### On the local structure of Deligne-Mumford stacks

Is it true that for any DM stack $\mathcal{X}$ (quasicompact, separated, of finite type over a field) there is a Zariski covering $\mathcal{U}_i \to \mathcal{X}$ by open substacks, such that for all ...

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### Extend a representation of a stabilizer group on a smooth DM stack to a locally free sheaf?

Consider a smooth tame Deligne-Mumford stack $[Y/G]$, a point $[p]$ on it with stabilizer group $H$. Is it true that every representation of $H$ can be extended to a locally free sheaf on $[Y/G]$?
...

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### Tensor Powers of 1-Dimensional Representations of a Finite Group

Let $G$ be a finite group acting on a commutative ring $R$ via ring maps. In doubt, one can assume $R$ to be noetherian or regular if one wants. Let $P$ be a $1$-dimensional free $R$-module with a ...

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### Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's ...

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### Diffeology as a sheaf on the site of smooth manifolds

Souriau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing ...

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### Zariski's Main Theorem for stacks

Let $X$ be a separated Deligne Mumford stack of finite type over a base field $k$ and $Y$ be a proper Deligne Mumford stack over $k$.
Assume there is a quasifinite, representable, surjective and ...

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### Pushout schemes/stacks

I would like to know in what generality do we have pushouts for schemes/stacks/derived stacks. More precisely, let $f : X \rightarrow Y$ be a proper flat surjective morphism of schemes of finite type ...

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### Stabilizer Action on vector bundle on a stack

Suppose you have a Deligne Mumford stack $X$ and a geometric point $x:Spec{k}\rightarrow X$ with stabilizer group $Stab(x)$.Let $F$ be a locally free sheaf on $X$.
How is the action of $Stab(x)$ on ...

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### Reference for Weighted Projective Stacks

For a sequence of positive integers $a_0, \ldots, a_n$ and a ring $R$, there is a graded ring $R[x_0,\ldots, x_n]$ where $x_i$ is in degree $a_i$. There is a corresponding $\mathbb{G}_m$-action on ...

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### Where does the ''hyper'' go?

There is a model structure on the category $Grpd$ of groupoids such that weak equivalences are equivalences of categories, cofibrations are the injections on objects (see nlab) and there is a Quillen ...

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### What is the difference between internal presheaves and presheaves on a total space?

Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$.
...

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### Chow ring of a $\mu_2$-gerbe

Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$?
(I assume they become ...

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### Why is there no stack of $\ell$-adic sheaves on a curve?

One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering such objects is that ...

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### Smooth map to the stack of G-bundles

Let $G$ a semisimple group and $B$ a Borel subgroup.
We denote by $Bun_{G}$ the stack of G-bundles.
Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$?
My question comes ...

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### KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...

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### Co-completeness of differential stacks?

I once heard a rumour that various nice categories of stacks were co-complete. Gepner and Henriques, working from the groupoids point of view, give a construction [link] of 2-colimits of topological ...

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### Rigidification and good moduli space (morphism) in the sense of Alper

Let $\mathcal{X}$ be an Artin stack. In "Abramovich, Graber, Vistoli - Twisted bundles and admissible covers", the authors describe a procedure to rigidify $\mathcal{X}$ by a central subgroup $H$ of ...

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### Uniqueness of the canonical etale coverings

This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata.
Let $X$ be a normal quasiprojective variety such that the canonical divisor $K_X$ is a ...

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### are moduli stacks deligne-mumford stacks in general

Let M be your favorite moduli stack over the field of complex numbers.
Is it reasonable to expect M to be a Deligne-Mumford stack?
I know this is true for the moduli space of curves of genus g, ...

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### How does the machinery of left-exact comonads generalize from sheaves to stacks?

Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...

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### universal families and maps to quotient stacks

Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductive group. Roughly ...

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### Finite-type Artin Stack over $\mathbb C$

Suppose I have an Artin stack $\mathfrak M$ locally of finite-type over $\mathbb C$ with presentation $M\rightarrow \mathfrak M$. Suppose further that $\mathfrak M$ "represents" (in the stack sense) ...

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### Geometric description of the Deligne-Mumford stacks

It is well known that a one-dimensional smooth Deligne-Mumford stack (over $\mathbb{C}$) could be described as a collection of its "stacky" points (finitely many) on its coarse moduli space with the ...

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### Passage from the moduli functor to the functor of points of the coarse moduli space

Let $F: (Sch)^{o}\to (Set)$ be a functor that admits a coarse moduli $Y$ (a scheme). We can consider $Y$ as a representable functor $h_{Y}: (Sch)^{o}\to (Set)$. Is there a direct way to produce $h_Y$ ...

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### What is the structure of the stack of complexes supported in dimension less than r?

Let $X$ be something. (smooth and projective variety over C are my assumptions)
The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...

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### Definition of derived category of a stack

In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows:
Let $X$ be a say complex variety with an action by an algebraic group $G$.
...

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### Twisting an object P by an H-Torsor I

I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.
The Statement
Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism ...

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### The Grothendieck Ring of Higher Stacks

The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that $[X] = [U] + ...

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### How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...

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### examples of moduli functors for which coarse moduli space does not exists

Well, the title almost says it all. I would like to list as many examples as possible of moduli functors, for which a coarse moduli space does not exist (and maybe explain why). So, examples such as ...

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### universal property of blow up for stacks?

I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is to be lifted to ...

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### When does sheaf cohomology commute with arbitrary direct sums?

It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map
...

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### On the local structure of stacks

1) Is it true that any Deligne-Mumford stack is locally a quotient stack $[X/G]$ with a finite group $G$?
2) Is it true that any Deligne-Mumford stack can be globally presented as a quotient stack ...

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### On the coarse moduli space of a stack

Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibred in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects of $\mathcal{X}^s$ ...