Čech cohomology

Sheaf cohomology defined by injective resolutions is conceptually clean but impossible to compute by hand. Cech cohomology is the down-to-earth, combinatorial recipe that actually lets you calculate: pick an open cover of your space, and build cohomology directly out of the sections of your sheaf on the cover sets and all their multiple overlaps. It turns a cohomology computation into bookkeeping with finite (or at least concrete) intersections, and on the spaces of algebraic geometry it usually reproduces the abstract sheaf cohomology exactly.

Precisely, fix an open cover U = {U_i} of X and a sheaf F. A Cech p-cochain assigns to every (p+1)-fold intersection U_{i_0} intersect ... intersect U_{i_p} a section of F there; these form a group C^p(U, F). The Cech differential d: C^p -> C^{p+1} is the alternating sum of restrictions, (d s)_{i_0 ... i_{p+1}} = sum_k (-1)^k s_{i_0 ... (omit i_k) ... i_{p+1}} restricted to the bigger intersection. Then d composed with d = 0, and the Cech cohomology H^p(U, F) is the cohomology of this complex; the full Cech cohomology H^p(X, F) is the direct limit over refinements of the cover. The low degrees are transparent: H^0 = global sections (a 0-cochain with zero differential is a family of local sections agreeing on overlaps, i.e. a global section by the sheaf axiom), and H^1 classifies the obstruction to gluing — a 1-cocycle is exactly a system of overlap data, like the transition functions of a line bundle, modulo those that come from a 0-cochain.

Cech cohomology is the standard computational engine: the cohomology of O(d) on projective space, hence Serre duality and Riemann-Roch in practice, is computed with the Cech complex of the standard affine cover. The decisive comparison theorem (Leray) says that if the cover is acyclic for F — every finite intersection of cover sets has vanishing higher cohomology of F (for quasi-coherent sheaves, finite intersections of affines are affine, hence acyclic) — then Cech cohomology relative to that cover equals the derived-functor sheaf cohomology. Honest cautions. First, for a BAD cover Cech cohomology can give the wrong answer; you must use an acyclic (Leray) cover, and only the limit over all refinements is guaranteed to agree in general. Second, even the refined Cech cohomology can differ from derived-functor cohomology on pathological (non-paracompact) spaces; the agreement is a theorem with hypotheses, cleanest for quasi-coherent sheaves on separated schemes and for paracompact spaces. So Cech is the tool you compute with, but 'Cech = sheaf cohomology' is a statement you must earn, not assume.