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5
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0answers
179 views

Homology of stack points (from math.stackexchange)

(This question is duplicated here) This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the ...
5
votes
0answers
172 views

Is $\mathcal{D} \bigl( \mathrm{QCoh}(\mathfrak{X}) \bigr)$ compactly generated?

An object $E$ in a triangulated category $\mathcal{T}$ with (small) coproducts is called compact if the functor $\mathrm{Hom}_{\mathcal{T}}(E,-)$ commutes with arbitrary coproducts or, equivalently, ...
4
votes
0answers
205 views

Coarse moduli spaces of quotient stacks

Suppose you have a separated Deligne Mumford quotient stack $[V/G]$ over a field of characteristic $0$, where $V$ is a quasiprojective variety and $G$ is an algebraic group that does not necessarily ...
4
votes
0answers
311 views

A question on an intuitive way to look at stacks

I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...
3
votes
0answers
105 views

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 ...
3
votes
0answers
231 views

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 ...
3
votes
0answers
167 views

Stack theoretic image?

If $X \xrightarrow{f} Y$ is a morphism of schemes then the scheme theoretic image of $f$ is the smallest closed subscheme $Z \subset Y$ through which $f$ factors through. Is this notion defined for ...
3
votes
0answers
90 views

Normalization of quotient stacks

Suppose you have a Deligne Mumford stack which is a quotient $[X/G]$ of a scheme $X$ by an algebraic group $G$ . What is the normalization of that? Is it true that its normalization is a quotient ...
3
votes
0answers
141 views

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 ...
3
votes
0answers
56 views

Are two “nice” transformation groupoids with the same coarse moduli and isomorphic inertia isomorphic?

Hi! I am stuck with the following question: suppose we have a semisimple connected algebraic group acting on a quasi-affine variety X by closed orbits, and suppose that inertia is flat. Suppose we ...
3
votes
0answers
188 views

Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks

Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$ Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that ...
2
votes
0answers
50 views

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 ...
2
votes
0answers
77 views

Subbundles of conformal blocks bundles over $\overline{\mathcal{M}}_{g,n}$

Let $\mathfrak{g}$ be a semisimple split Lie algebra over a field $k$ of characteristic zero. Let $l$ be a non-negative integer and let $P_{l}^{n}$ be the set of $n$-tuples of dominant integral ...
2
votes
0answers
103 views

Are there any results on stable maps to Artin stacks with infinite stabilizers?

The Abramovich-Vistoli/Chen-Ruan theory of twisted stable maps into Deligne-Mumford stacks is extremely useful, as is the generalization to tame Artin stacks in positive characteristic. I am ...
2
votes
0answers
153 views

Subgroups of a group algebraic space

I found in the literature many references on the representability of quotients of group schemes but almost nothing about subgroups. For this reason I hope that my question is a silly one and that what ...
2
votes
0answers
177 views

Lang isogeny for group stacks

Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
1
vote
0answers
138 views

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 ...
1
vote
0answers
216 views

Fourier-Mukai transforms on stacks

I have "generic" questions about Fourier-Mukai transforms. Question 1: Does there exist a well-defined notion of Fourier-Mukai transform on (Deligne-Mumford) stacks? Question 2: Do there exist ...
0
votes
0answers
81 views

extensions of vector bundles on algebraic stacks

Let $U\subset X$ be an open immersion of separated algebraic stacks of finite type over a field , Let $E$ be a vector bundle on $U$ . In which cases can we extend $E$ to a vector bundle on $X$? Same ...