Grothendieck spectral sequence: Difference between revisions

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In [[algebraic topology]], the '''De Rham–Weil theorem''' allows computation of [[sheaf cohomology]] using an [[acyclic sheaf|acyclic]] [[injective resolution|resolution]] of the sheaf in question.
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Let <math>\mathcal F</math> be a [[sheaf (mathematics)|sheaf]] on a [[topological space]] <math>X</math> and <math>\mathcal F^\bullet</math> a resolution of  <math>\mathcal F</math> by acyclic sheaves. Then 
 
:<math> H^q(X,\mathcal F) \cong H^q(\mathcal F^\bullet(X)), </math>
 
where <math>H^q(X,\mathcal F)</math> denotes the <math>q</math>-th [[sheaf cohomology]] [[Inverse (mathematics)|group]] of <math>X</math> with coefficients in  <math>\mathcal F.</math>
 
The De Rham–Weil theorem follows from the more general fact that derived functors may be computed using acyclic resolutions instead of simply injective resolutions.
 
{{PlanetMath attribution|id=6333|title=De Rham-Weil theorem}}
 
{{DEFAULTSORT:De Rham-Weil theorem}}
[[Category:Homological algebra]]
[[Category:Sheaf theory]]

Latest revision as of 13:04, 5 May 2014

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