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 ${\displaystyle {\mathcal {F}}}$ be a sheaf on a topological space ${\displaystyle X}$ and ${\displaystyle {\mathcal {U}}}$ an open cover of ${\displaystyle X.}$ If ${\displaystyle {\mathcal {F}}}$ is acyclic on every finite intersection of elements of ${\displaystyle {\mathcal {U}}}$ (meaning that ${\displaystyle H^{i}(U_{1}\cap \dots \cap U_{p},{\mathcal {F}})=0}$ for all ${\displaystyle i\geq 1}$ and all ${\displaystyle U_{1},\dots ,U_{p}\in {\mathcal {U}})}$, then

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

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

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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