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."

This article incorporates material from Leray's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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