Leray's Theorem

In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with ÄŒech 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 $H^i(U_1 \cap \dots \cap U_p, \mathcal) = 0$ for all $i \ge 1$ and all $U_1, \dots, U_p \in \mathcal)$, 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 ÄŒech 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."

Sheaf theory
Theorems in algebraic geometry
Theorems in algebraic topology

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