Counting global sections, measuring obstructions, and the most powerful theorem in algebraic geometry
Sheaf cohomology measures the failure of local-to-global constructions. The global sections H0(X, ℱ) of a sheaf ℱ are easy to define, but the interesting question is: when can compatible local sections be glued into a global one? The higher cohomology groups Hi(X, ℱ) measure exactly the obstructions to gluing. On affine schemes, all higher cohomology vanishes — this is Serre's theorem — so cohomology is a genuinely global phenomenon.
The crown jewel is the Riemann-Roch theorem, which computes the Euler characteristic χ(L) = ∑(−1)i dim Hi(X, L) of a line bundle L on a curve. For a smooth projective curve of genus g, the formula is χ(L) = deg(L) + 1 − g. Combined with Serre duality Hi(X, ℱ) ≅ Hn−i(X, ωX ⊗ ℱ∨)∨, this gives precise control over the dimensions of spaces of sections.
Čech cohomology provides a concrete way to compute sheaf cohomology using open covers. Given an open cover {Ui} of X and a sheaf ℱ, a Čech 0-cochain is a choice of section si ∈ ℱ(Ui) for each i. It is a cocycle if si and sj agree on overlaps Ui ∩ Uj, and a coboundary if it comes from a global section. The group H1 = cocycles/coboundaries measures exactly the obstructions to gluing local sections into a global one.
On ℙ1, the line bundle 𝒪(n) has global sections that are homogeneous polynomials of degree n. For n ≥ 0, dim H0(ℙ1, 𝒪(n)) = n + 1 and H1 vanishes. For n < 0, there are no global sections, but H1 is nonzero — the obstruction to gluing reflects the "twisting" of the bundle.
For a line bundle L on a smooth projective curve C of genus g, the Riemann-Roch theorem states χ(L) = dim H0(C, L) − dim H1(C, L) = deg(L) + 1 − g. This single formula connects topology (the genus), algebra (the degree of the divisor), and analysis (the dimension of the space of sections). It is one of the most powerful tools in all of algebraic geometry.
Serre duality provides the symmetry that completes the picture: H1(C, L) ≅ H0(C, ωC ⊗ L∨)∨, where ωC is the canonical bundle (the sheaf of differentials). This duality turns H1computations into H0 computations on a "dual" line bundle, often making intractable problems solvable. In higher dimensions, Serre duality generalizes to pair Hi with Hn−i.