Leray's theorem

Template:Short description In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology.

Let $ℱ$ be a sheaf on a topological space $X$ and $𝒰$ an open cover of $X.$ If $ℱ$ is acyclic on every finite intersection of elements of $𝒰$ (meaning that $H^i(U_1\cap \dots \cap U_p,ℱ)=0$ for all $i\ge 1$ and all $U_1,\dots ,U_p\in 𝒰)$, then

${\check{H}}^q(𝒰,ℱ)=H^q(X,ℱ),$

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

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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