All Questions
11 questions with no upvoted or accepted answers
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Idea of presheaf cohomology vs. sheaf cohomology
Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech ...
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5
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About the left adjoint of $f^*$
In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ...
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4
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Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one
I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
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Sheaf-type property for Derived Categories?
Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...
- 1,761
4
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Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
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3
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Image of Obstruction Map for Relative Quot-scheme
Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ ...
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3
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What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?
It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\...
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details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
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Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
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2
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On the generalization of a Cech-to-sheaf type spectral sequence
Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
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How to deduce Künneth from its relative version (in cohomology of sheaves)
Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism
$$f_!(M\boxtimes N)=p_! M\otimes q_!N$$
in the derived category of "sheaves" over $S$, where ...
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