PlanetMath: resolution of a sheaf

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resolution of a sheaf

(Definition)

Given a sheaf $\mathcal F$ on a topological space and a pair $(\mathcal F^\bullet, d^\bullet)$ where $\mathcal F^\bullet = \{\mathcal F^q\}_{q\in\mathbb{N}}$ is a family of sheaves on and $d^\bullet=\{d^q\}_{q\in\mathbb{N}}$ is a family of sheaf morphisms

$\displaystyle d^q\colon\mathcal F^q\to\mathcal F^{q+1},$

we say that $(\mathcal F^\bullet, d^\bullet)$ is a resolution of $\mathcal F$ if there exists an injection $j\colon\mathcal F\to\mathcal F^0$ such that the sequence

$\displaystyle 0\to\mathcal F\overset{j}\to\mathcal F^0\overset{d^0}\to\mathcal F^1\overset{d_1}\to\cdots\to\mathcal F^q\to\dots$

is exact.

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"resolution of a sheaf" is owned by Simone. [ full author list (3) ]

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See Also: De Rham-Weil theorem

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Cross-references: sequence, injection, morphisms, sheaves, topological space, sheaf
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This is version 5 of resolution of a sheaf, born on 2004-10-09, modified 2007-01-27.
Object id is 6334, canonical name is ResolutionOfASheaf.
Accessed 1161 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

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