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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)

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. Reference: Wikipedia.

This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants.

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Proposition 2.7.2 from Kashiwara and Schapira, Sheaves on Manifolds

I am reading Kashiwara-Schapira's Sheaves on manifolds, and I am having trouble understanding their proof of Proposition 2.7.2 which says the following : Let $X$ be a Hausdorff space, $F \in D^+(\mathbb{Z}_X)$, and let $\{U_t\}_{t\in \mathbb{R}}$…
stratified
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Cup product of Cech cohomology graded commutative

Let $(X,\mathscr{O}_X)$ be a ringed space. Let $\{U_i\}_{i\in I}$ be a finite open cover of $X$. Let $U_{i_0\cdots i_n}=\cap_{i_j}U_{i_j}$. Consider the Cech cohomology given by the this cover, $\check{H}^j(\{U_i\},\mathscr{O}_X)$. For general…
Atticus Christensen
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Let $S^1$ be the circle (with its usual topology), and let $\mathbb{Z}$ be the constant sheaf. Show that $H^1(S^1,\mathbb{Z})\cong\mathbb{Z}$.

Rather than be given a proof, I'd like to know why my proof is wrong. (It's clearly wrong because I got the wrong final answer.) We build an injective resolution of $\mathbb{Z}$; one possibility is $$…
Erik
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Interpreting the fist Cech cohomology group

Given a presehaf of abelian groups $\mathcal{F}$ and an open cover $\mathcal{U}:=\{U_i\}_{i=0}^n$, I can define the Cech cohomology groups $\check{H}^q(\mathcal{U},\mathcal{F})$. It s well known that elements of…
John
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Question about cohomology of differential sheaf.

In $\mathbb{P}^1_{\mathbb{C}}$, given affine covers $U=(v\neq0),V=(u\neq0)$, we write element $(a,b)$ of $C^0(\{U,V\},\Omega^1)=\{(a,b): a\in \Omega^1(U), b\in \Omega^1(V)\}$ as $a=\sum^{\infty}_{n=0}a_n u^ndu$, $b=\sum^{\infty}_{n=0}b_n…
Yuyi Zhang
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Čech cohomology of fine sheaves

Let $X$ be a topological space with open cover $\mathfrak U$ and abelian sheaf $\mathcal F$. We know that if $\mathcal F$ is flasque, then higher Čech cohomology with respect to the covering $\mathfrak U$ vanishes, i.e. $\check{H}^p(\mathfrak U,…
johnnycrab
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Induced map from sheaves into cohomologies?

Suppose I have an short exact sequence of sheaves: $$0 \to \mathcal{E} \to \mathcal{F} \to \mathcal{G} \to 0$$ My book says that this induces a long exact sequence of cohomologies (by snake lemma): $$0 \to H^0(X,\mathcal{E)} \to H^0(X,\mathcal{F})…
mtheorylord
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inverse image with compact support

In several notes, when introducing operator $f^{!}$ (as an adjoint of $f_!$, actually $Rf_!$), we have to pass to derived category. I am wondering what the reason is. (Manin and Gelfand's homological algebra book explains it briefly (say Page 229)…
user72443
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problems to the Dolbeault cohomology

I am currently reading the section on the Dolbeault cohomology in Griffiths & Harris Principles of Algebraic Geometry and am having trouble understanding some problems. It is obvious that $${ H }^{ q }\left( \mathbb{ C }^{ n },\mathcal{O} \right)…
unicornki
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Does permuting indices of a cocycle leave the Cech cohomology class the same?

Consider a space $X$, $\{U_i\}_i$ an open covering, and $\mathcal{F}$ a sheaf on $X$. Consider $(c_{i_0,\ldots,i_n})$ a Cech $n$-cocycle. It is theorem that every Cech $n$-cocyle is cohomologous to a an alternating one, ie a cocycle,…
Atticus Christensen
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Čech cohomology - independence of covering

Let $X$ be a topological space and let $\mathcal{F}$ be a sheaf over $X$. Suppose we have a open covering $\mathcal{U}$ of $X$. When $\check{\mathrm{H}}^n(X,\mathcal{F})\cong \check{\mathrm{H}}^n(\mathcal{U},\mathcal{F}) $ ? (When Čech cohomology…
mikis
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References on $\operatorname{Ext}$ of sheaves and base change.

Let $X\to S$ be a proper morphism of Noetherian schemes. Let $\mathcal E,\mathcal F$ be coherent $\mathcal O_X$-modules, flat over $S$. Let $g:T\to S$ be a morphism of coherent schemes. There is a base change map $$g^*\operatorname{Ext}_X^i(\mathcal…
Display Name
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Quasi-isomorphism of complexes of sheaves

I am currently reading Robert Friedman's notes on sheaf cohomology and hypercohomology http://www.math.columbia.edu/~rf/cohomology.pdf In it he defines quasi-isomorphism of complexes of sheaves by saying the $i^{th}$ cohomology sheaves…
amd1234
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Non constant sheaf on a contractible space

while trying to calculate the coohomology (with complex coefficients) of a bundle over a simply connected base, I met the following PROBLEM Suppose you have a space $X$ which contracts on $x \in X$, and a sheaf $F$ on $X$ such that $F_p \simeq V$…
frame95
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