PlanetMath: Čech cohomology group

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Čech cohomology group

(Definition)

Let $\mathcal F$ be a sheaf of abelian groups on a topological space and consider an open covering $\mathcal U=\{U_i\}_{i\in I}$ of . For the sake of simplicity denote

$\displaystyle U_{i_0i_1\cdots i_q}=U_{i_0}\cap U_{i_1}\cap\dots\cap U_{i_q}.$

The group $\check C^q(\mathcal U,\mathcal F)$ of Čech

-cochains is the set of families

$\displaystyle c=(c_{i_0i_1\cdots i_q})\in\prod_{(i_0,\dots,i_q)\in I^{q+1}}\mathcal F(U_{i_0i_1\cdots i_q}).$

The group structure on $\check C^q(\mathcal U,\mathcal F)$ is the obvious one deduced from the addition law on sections of $\mathcal F$ .

The Čech differential

$\displaystyle \delta^q\colon\check C^q(\mathcal U,\mathcal F)\to\check C^{q+1}(\mathcal U,\mathcal F)$

is defined by the formula

$\displaystyle (\delta^q c)_{i_0\cdots i_{q+1}}=\sum_{0\le j\le q+1}(-1)^j c_{i_0\cdots\widehat{i_j}\cdots i_{q+1}}\vert _{U_{i_0\cdots i_{q+1}}},$

and we set $\check C^{q}(\mathcal U,\mathcal F)=0$ , $\delta^q=0$ for

. Easy computations show that $\delta^{q+1}\circ\delta^q=0$ . We get therefore a cochain complex $(\check C^\bullet(\mathcal U,\mathcal F),\delta)$ , called the complex of Čech cochains relative to the covering $\mathcal U$ .

The -th Čech cohomology group of $\mathcal F$ relative to $\mathcal U$ is

$\displaystyle \check H^q(\mathcal U,\mathcal F)=H^q(\check C^\bullet(\mathcal U,\mathcal F)).$

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"Čech cohomology group" is owned by Simone. [ full author list (2) ]

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Other names:

Cech cohomology group

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Cross-references: cohomology group, complex, cochain complex, sections, addition, obvious, group, topological space, abelian groups, sheaf
There are 3 references to this entry.

This is version 2 of Čech cohomology group, born on 2004-10-10, modified 2007-01-27.
Object id is 6346, canonical name is CechCohomologyGroup2.
Accessed 2780 times total.

Classification:

AMS MSC:

18G60 (Category theory; homological algebra :: Homological algebra :: Other homology theories)

Pending Errata and Addenda

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