All Questions
Tagged with algebraic-stacks derived-categories
9 questions
3
votes
0
answers
197
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Concrete reasons to study derived categories of quasi-coherent sheaves on algebraic stacks?
Title says it all. There seems to be a lot of work done on derived categories of quasi-coherent sheaves on stacks, and yet I couldn't find many "external" applications. I'm mainly wondering ...
- 143
5
votes
1
answer
327
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Comparison between pushforward-pullback and quasi-coherent pushforward-pullback
In the following, an algebraic stack means a stack over the big fppf site of a scheme, admitting a smooth, representable, surjective morphism from a scheme (no separation hypotheses), and let $\mathsf{...
- 1,349
7
votes
1
answer
628
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Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,349
1
vote
1
answer
235
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Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface
Let $k$ be an algebraically closed field of $\text{ch}(k) =0$.
Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack.
Let $\mathbf{P}(1,1,2)$ be the weighted ...
- 13
2
votes
0
answers
174
views
$D^b_\text{Coh}(X)$ for a smooth, proper Deligne-Mumford stack
Let $X$ be a smooth, proper DM stack over a field $k$. I see in Hall-Rydyh's paper "Perfect Complexes on Algebraic Stacks" (https://arxiv.org/abs/1405.1887) a discussion of compact ...
- 511
3
votes
0
answers
123
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Commutative group stacks and Galois cohomology
"Classically", if we consider an abelian variety $A$ over some number field $k$, we get a $Gal(\bar{k}/k)$-module $A(\bar{k})$, or equivalently a sheaf of abelian groups on the étale site $\...
- 4,386
4
votes
0
answers
315
views
Skyscraper sheaf on a stack associated to a singular surface
Suppose $X$ is a normal projective surface with a du Val singularity. In this case, we know a crepant resolution $Y$ exists, and results of Kawamata (https://arxiv.org/abs/0804.3150, Corollary 3.5) ...
- 153
4
votes
1
answer
404
views
Derived pullback of the coarse moduli morphism
Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^*$ is fully faithful and induces an ...
- 252
14
votes
2
answers
1k
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, ...
- 251