Today’s main goal is the Leray theorem (though at the end I have to ask a question):

Theorem 1Let be a sheaf on , and an open cover of . Supposefor all -tuples , and all . Then the canonical morphism

is an isomorphism for all .

This seems rather useless, because the theorem presupposes the vanishing of (regular) cohomology on the covering. However, in many cases it turns out to be helpful. If is a separated scheme, an open affine cover of , and quasi-coherent, it applies. The reason is that each of the intersections are all affine by separatedness, so has no cohomology on them by a basic property of quasi-coherent sheaves. This gives a practical way of computing sheaf cohomology in algebraic geometry. Hartshorne uses it to compute the cohomology of line bundles on projective space.

Another instance arises when is the sheaf of holomorphic functions over some Riemann surface . In this case is a covering of charts. It is a theorem (which I will eventually prove) that for any open subset of (which any intersection of the ‘s is isomorphic to), the sheaf has trivial cohomology. (more…)