Leray's theorem

Short description: Relates abstract sheaf cohomology with Čech cohomology

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 ${𝒰}$, then

${\check{H}}^q({𝒰},{ℱ})={\check{H}}^q(X,{ℱ}),$

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."