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 [math]\displaystyle{ \mathcal F }[/math] be a sheaf on a topological space [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \mathcal U }[/math] an open cover of [math]\displaystyle{ X. }[/math] If [math]\displaystyle{ \mathcal F }[/math] is acyclic on every finite intersection of elements of [math]\displaystyle{ \mathcal U }[/math], then

[math]\displaystyle{ \check H^q(\mathcal U,\mathcal F)= \check H^q(X,\mathcal F), }[/math]

where [math]\displaystyle{ \check H^q(\mathcal U,\mathcal F) }[/math] is the [math]\displaystyle{ q }[/math]-th Čech cohomology group of [math]\displaystyle{ \mathcal F }[/math] with respect to the open cover [math]\displaystyle{ \mathcal U. }[/math]

References

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