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…)