# Questions tagged [sheaf-cohomology]

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. (Def: http://en.m.wikipedia.org/wiki/Sheaf_cohomology)

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### fppf/ etale Cohomology calculate with Cech cohomology

Let $R$ be a commutative ring with one (so living in standard commutative algebra setting) and let $\phi: R \to S$ a faithfully flat. Then the so called Amitsur complex $R \to S^{\otimes \bullet +1}$ ...
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### Construction of Quot schemes and construction of certain closed subschemes

I am still studying Nitsure's Construction of Hilbert and Quot schemes, and I am yet again stuck, this time on page 25. I understand what lemma 5.4 says, but I do not get its proof. Let me sketch the ...
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### Injective map between sheaf cohomologies

I am currently working through Forster's Lectures on Riemann Surfaces and I may be on the right track on this exercise, but I'd like to make sure. The exercise in question is 12.3: Let $X$ be a ...
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### Semi-continuity theorem and construction of Hilbert and Quot schemes

I am studying Nitsure's wonderful essay Construction of Hilbert and Quot schemes, and I am stuck on page 21. Given a Noetherian schemes $S$ and a coherent sheaf $\mathcal{F}$ over $\mathbb{P}^{n}_{S}$,...
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### Vanishing of the Zariski cohomology $H^n(X,\mathbb{Z}[\mathbb{G}_m])$ for $n>1$

Let $X$ be a smooth and irreducible variety over a field. Does the Zariski cohomology $H^n(X,\mathbb{Z}[\mathbb{G}_m])$ vanish for $n>1$? Here $\mathbb{Z}[-]\colon Set\to Ab$ is (the Zariski ...
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### sheaf and de Rham cohomology of projective lines glued to order $n$

Let $X = \mathbb{P}^1_k \cup_n \mathbb{P}^1_k$ be the union of two projective lines, glued together at a single point, where the gluing is of order $n$. I would like to compute the sheaf cohomology ...
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### Meaning of certain cohomology classes and morphisms in Prop 3.3 of Deligne-Illusie's paper on mod $p^2$ liftings and decompos. of the de Rham complex

I am still struggling with Deligne and Illusie's paper (https://eudml.org/doc/143480). They say on page 261, in the course of the proof of theorem 3.3: The class $e(K)$ (which is associated to the ...
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### Čech cohomology of sites: How does a “composite” morphism induce an element of $\operatorname{Ext}^{2}$?

This is yet another question concerning Deligne and Illusie's paper on $W_{2}(k)$ liftings and the degeneration of the Hodge-de Rham spectral sequence (https://eudml.org/doc/143480). In the proof of ...
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### Étale cohomology vs flat cohomology

I'm recently reading the Milne's Etale Cohomology (1980 Princeton University Press). When I read the part of etale cohomology vs flat cohomology in page 115, I have trouble understanding the first ...
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### How does a morphism in a distinguished triangle induce an element of $\operatorname{Ext}^{2}$

On p. 260 of Deligne and Illusie's paper (https://eudml.org/doc/143480) it says: Let $e(K)\in\operatorname{Ext}^{2}(H^{1}(K),H^{0}(K))$ be the class defined by the degree $1$ morphism in the ...
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### Cohomology(ies) of simplicial sheaves

Let $X$ be a topological space and denote by $Ab(X)$ the category of abelian sheaves on $X$. My question is on the category of simplicial abelian sheaves $[\Delta^{op},Ab(X)]$. A natural way to define ...
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### Representable cohomology theories in motivic homotopy theory

I am reading Mazza, Voevodskys and Weibels book on Lecture Notes on Motivic Cohomology and have grown curious about the following question: Which cohomology theories on $Sm/k$ is representable, i.e. ...
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### Hodge star operator and “Serre duality”

I am familiar with the Hodge star operator or Hodge duality in the theory of finite-dimensional differentiable manifolds, which gives an isomorphism $\star:\Omega^{i}(M)\longrightarrow\Omega^{n-i}(M)$ ...
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### Liu Exercise 5.21

I'm trying to solve Liu's exercise 2.1 Chapter 5 which states: Here is what I'm thinking so far: a) This is vacuously true since J is a single element set so there is no $i<j$ and this is trivial. ...
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### Proving that the Čech complex is actually a complex

All authors I have seen handwave the proof of the Čech complex actually being a complex as a calculation. But I tried to do it and I don't see how it works. Maybe there's a mistake in my calculation. ...
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### Hom, H$^n$ and tensor product
If $X$ is a scheme, $\mathcal{F}$ is a quasi-coherent sheaf and $L$ is an ample line bundle, why it holds $H^0(X, \mathcal{F} \otimes L^n) = Hom(L^{-n}, \mathcal{F})$?