If every nonempty finite intersection of members of an open cover has vanishing higher sheaf cohomology for , then the cover's Čech cohomology computes . An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.
If is a closed immersion of Noetherian schemes, is separated, and is quasi-coherent on , thenIndeed, an affine cover of induces an affine cover of , and the two associated Čech cochain complexes are termwise identical.
If a separated scheme has an acyclic affine cover by open sets, then every quasi-coherent sheaf on it has vanishing sheaf cohomology in degrees at least . The corresponding Čech cochain complex has no terms in those degrees.
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Leray's theorem, often referred to in the context of topology or functional analysis, generally pertains to the existence of solutions for certain types of partial differential equations (PDEs) or, more broadly, variational problems. One of the prominent formulations of Leray's theorem deals with the existence of weak solutions for the Navier-Stokes equations, which describe the motion of fluid substances.