Leray's theorem

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In algebraic geometry, Leray's theorem relates abstract sheaf cohomology with Čech cohomology.

Let \mathcal F be a sheaf on a topological space X and \mathcal U an open cover of X. If \mathcal F is acyclic on every finite intersection of elements of \mathcal U, then

 \check H^q(\mathcal U,\mathcal F)= H^q(X,\mathcal F),

where \check H^q(\mathcal U,\mathcal F) is the q-th Čech cohomology group of \mathcal F with respect to the open cover \mathcal U.


  • Bonavero, Laurent. Cohomology of Line Bundles on Toric Varieties, Vanishing Theorems. Lectures 16-17 from "Summer School 2000: Geometry of Toric Varieties."

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