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

Theorem 1 Let {\mathcal{F}} be a sheaf on {X}, and {\mathfrak{U} = \{ U_i, i \in I\}} an open cover of {X}. Suppose\displaystyle H^n( U_{i_1} \cap \dots \cap U_{i_k}, \mathcal{F}|_{ U_{i_1} \cap \dots \cap U_{i_k}}) = 0

for all {k}-tuples {i_1, \dots , i_k \in I}, and all {n>0}. Then the canonical morphism\displaystyle H^n( \mathfrak{U}, \mathcal{F}) \rightarrow H^n( X, \mathcal{F})

is an isomorphism for all {n}. 

 

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 {X} is a separated scheme, {U_i} an open affine cover of {X}, and {\mathcal{F}} quasi-coherent, it applies. The reason is that each of the intersections { U_{i_1} \cap \dots \cap U_{i_k}} are all affine by separatedness, so {\mathcal{F}} 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 {\mathcal{O}} is the sheaf of holomorphic functions over some Riemann surface {X}. In this case {\{U_i\}} is a covering of charts. It is a theorem (which I will eventually prove) that for any open subset of {\mathbb{C}} (which any intersection of the {U_i}‘s is isomorphic to), the sheaf {\mathcal{O}} has trivial cohomology. (more…)