Leray's theorem

In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) 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 ${\mathcal {U}}$ (meaning that $H^{i}(U_{1}\cap \dots \cap U_{p},{\mathcal {F}})=0$ for all $i\geq 1$ and all $U_{1},\dots ,U_{p}\in {\mathcal {U}})$, 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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