Questions tagged [algebraic-stacks]
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284
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Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12k
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Enough injectives in the category of quasi-coherent sheaves on a stack
For a scheme $X$, I have a reference - https://stacks.math.columbia.edu/tag/077P - that says there are enough injectives in the category $\text{QCoh}(X)$. I am looking for a reference that says the ...
- 966
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Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?
I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$.
Recently I was pointed to Katz and Mazur's book, ...
- 34.8k
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References for orbifold curves
I am looking for a good reference (if there is any) for the theory of orbifold curves from the perspective of stacks. By an orbifold curve I mean something like a $1$-dimensional irreducible Deligne-...
- 341
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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\...
- 1,179
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Classifying stack for finite flat group scheme
Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...
- 435
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Is a finite morphism of Deligne-Mumford stacks proper?
The situation that I am in is the following. Let $\mathcal{X}$ be a smooth Deligne-Mumford stack over a field $k$. Let $X$ be a $k$-scheme together with a morphism $\pi;\mathcal{X}\rightarrow X$ (you ...
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What are the categories of IND and PRO schemes?
below is a mathexchange question with no answers so I drop it here.
I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are, in particualer the relations with ...
- 1,125
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Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks
I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...
- 737
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Relationship between $\infty$-categories of ind-coherent sheaves on the base and total space
My question is vaguely as follows, let $G\to E\to X$ be a principal $\infty$-bundle for some group object $G$ in a category of ($\infty$-)stacks. Can we recover the category $\operatorname{IndCoh}(E)$ ...
- 1,294
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Finite generation of stack cohomology
Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra?
For instance, $\text{H}^*(\text{B}\mathbf{G}...
- 5,565
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Dualizing sheaf for classifying stack and duality
For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...
- 752
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Hypercover and hyper descent
I am trying to understand the descent condition using hypercovers. The condition says that a hyper cover of a scheme $X$ is a simplicial set $Y_{\bullet}$ that satisfies the condition $Y_n\rightarrow ...
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Moduli stack of l-adic sheaves?
Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors.
Let $\ell$ be a prime ...
- 341
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Grothendieck purity for Brauer groups of stacks
Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
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