Leray's theorem

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 $\mathcal U.$

References

• 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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