Leray's theorem

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}}} , 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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