# Tagged Questions

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### How should a homotopy theorist think about sheaf cohomology?

As a student of homotopy theory or algebraic topology, I have a certain outlook as to how one ought to think of a cohomology theory. There are axioms that help us with rudimentary computations, there ...
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### Cohomologically trivial stacks

The following theorem of Serre is well-known: A noetherian scheme $X$ is affine if and only if $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$. (...
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### How to Draw Complex Line Bundles

I am giving a presentation soon on the Classification of Complex Line Bundles and I would like to have some very "basic" visualizations to use as examples. Background and Context I am considering ...
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### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...
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### Where am I suppose to actually learn how to compute hypercohomology?

I'm reading about algebraic de Rham cohomology over characteristic zero which is constructed using hypercohomology. Already, constructing injective resolutions is difficult, and coupling this with ...
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### Eilenberg-Steenrod axioms of sheaf cohomology

Cohomology of a space is often defined axiomatically: a cohomology theory is a functor from pairs of spaces to abelian groups satisfying the Eilenberg-Steenrod axioms. Is there a similar ...
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### Deducing properness from H^i(X,F) finitely generated over \Gamma(O_X)

Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is ...
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### Dualizing complex of the product of two locally compact spaces

Hello! In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
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### Cohomology of line bundles

For sure answers to my questions are well known - but I never saw them anywhere. Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Let $A_i$ be the ...
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### Local cohomology groups and linearity

I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of ...
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### Sheaf cohomology with compact supports (and Verdier duality?)

Consider a manifold and a complex where cochains are sections of vector bundles and coboundary maps are differential operators, which are locally exact except in lowest degree (think de Rham complex). ...
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### “Flat” and “Affine” morphisms of smooth manifolds

Let $f: X \to Y$ be a qcqs morphism of qcqs schemes. We have the following characterizations: $f$ is flat $\iff$ $f^*:QCoh(Y) \to QCoh(X)$ is exact. $f$ is affine $\iff$ $f_*: QCoh(X) \to QCoh(Y)$ ...
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### Basic properties of Nisnevich cohomology; $l'$-topology?

I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, ...
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### Do general sheaves on P^2 have cohomology governed by their Euler characteristic?

Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$. If $\xi$ has Euler characteristic 0, then apparently there is ...
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### Birational Invariants

Let $X$ be a smooth rational variety of dimension $n$. We have $\dim H^0(X,\Omega_X^p) = \dim H^0(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^p)$ for any $p$. These are Hodge numbers. I know that we can not ...
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### Sheaves with isomorphic cohomology, but not quasi-isomorphic

Suppose I have two (constructible) sheaves of vector spaces $F$ and $G$ over the same base space that have isomorphic cohomology (degree by degree), but no sheaf map inducing this isomorphism (i.e. ...
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### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer. Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...
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Euler Characteristic of Sheaves and the Generalized Gauss-Bonnet Theorem Consider a sheaf $\mathscr{F}$ over a site $\mathscr{(C,J)}$, where $\mathscr{C}$ is a small category with a coverage $\... 1answer 560 views ### Are injective quasi-coherent modules acyclic? Let$X$be a scheme and$F$be an injective object of$\mathrm{Qcoh}(X)$. Is it true that$F$is acyclic with respect to the usual sheaf cohomology? For noetherian schemes$X$this is well-known; ... 1answer 185 views ### Does the nearby cycle functor commute with the Verdier duality? I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference ... 1answer 408 views ### Construction of generalized Eilenberg-MacLane spaces The Eilenberg-MacLane spaces$K(G,q)$are readily generalized to study cohomology with local coefficients.The generalized Eilenberg-MacLane space$K_{\pi}(G,q)$are spaces with only two nnvanishing ... 1answer 204 views ### Surjectivity of certain cohomology groups on hypersurfaces of high degree I had been reading an article by Spencer Bloch. There is a remark in this text which states the surjectivity of a particular map between cohomology groups without explaining further. I had been trying ... 1answer 195 views ### A mysterious quasi-isomorphism in Kashiwara-Schapira's proof of HKR On p. 127 of Kashiwara-Schapira's paper "Deformation Quantization Modules", there is the following situation:$X$is a smooth complex (quasi?)projective variety and$\delta\colon X\to X\times X$is ... 1answer 167 views ### On push-forward of the constant sheaf for fibrations Let$f\colon E\to B$be a fiber bundle with a connected fiber$F$,$f$is proper. Let$\underline{\mathbb{C}}_E$be the constant sheaf on$E$. Let$f_*(\underline{\mathbb{C}}_E)$denote its direct ... 1answer 356 views ### Leray's theorem up to some degree I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology. The theorem states that if we have a space$X$, a sheaf$\mathcal{F}$and a covering of$X$such ... 1answer 211 views ### Relating deformations of a scheme to deformations of its singular locus Let$X$be a normal scheme with quotient singularities and$Y\subset X$its singular locus. The first order deformations of$X$are parametrized by$\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ... 0answers 50 views ### Reference request: local cohomology in disjoint union I have a topological space$X$and two disjoint, closed subspaces$Y$and$Z$of$X$. I believe that in this situation, for any abelian sheaf$\mathcal{F}$on$X$and any$p \in \mathbb{N}$, there is ... 0answers 142 views ### Proper base change for non-quasicoherent sheaves For a proper flat map$f: X \to Y$(of reasonable schemes) and a closed embedding$i: Y' \to Y$, we know by EGA the base change quasi-isomorphism, where$f'$and$i'$are the pullbacks of$f$and$i$: ... 0answers 108 views ### Cohomology of$Sym^m Q \otimes Sym^k Q \otimes L^p$Let$V$be a complex vector space. Let$L=\mathcal{O}(-1)$and$Q=V/L$be the quotient bundle over$\mathbb{P}V$. I'm trying to compute the cohomologies with coefficients in$Sym^m Q \otimes Sym^k Q \...
Let $X$ be a proper scheme over a henselian discrete valation ring. I have a Nisnevich sheaf $F$ of which has only one stalk at the generic point of $X$ (and all other stalks vanish). I believe that ...