Čech cohomology and sheaf cohomology

For an ordered open cover, a degree-p Čech cochain assigns a section to every increasing intersection of p+1 opens. The differential is the alternating sum of restrictions, omitting one index at a time. Each twice-omitted restriction appears with opposite signs, so the differential squares to zero. Degree-zero cocycles are exactly compatible families and hence global sections. A degree-one cocycle represents overlap data, while a coboundary is overlap data obtained by changing local choices on the individual opens.

If the sheaf is a line bundle, choose local frames and translate all overlap sections into the same frame before subtracting them. Forgetting a transition function can turn a correct gluing problem into the cohomology of the wrong sheaf. Signs do not change the span of coboundaries in our two-chart calculation, but powers of the coordinate do. Naming the frame convention explicitly makes the quotient calculation reproducible and prevents a positive twist from accidentally being treated as a negative one.

Derived sheaf cohomology is obtained by applying global sections to an injective resolution in the category of sheaves of modules. It is independent of the chosen resolution. For a quasi-coherent sheaf on an affine scheme, all positive-degree cohomology groups vanish. Consequently, if a cover has affine finite intersections, its Čech complex computes the derived sheaf cohomology of a quasi-coherent sheaf in every degree. This is the comparison theorem used here, and it requires both the sheaf condition and the acyclicity input.

A separated scheme has affine intersections of affine opens, so a quasi-compact separated scheme admits a finite affine cover suitable for this calculation. More general acyclic covers are possible; separatedness is a convenient sufficient condition rather than the definition of Čech cohomology. For an arbitrary sheaf or an arbitrary cover, higher cohomology on the intersections can contribute through a spectral sequence, and the naive Čech groups need not equal the derived groups. Local acyclicity must therefore be checked before interpreting a calculated quotient as a global invariant.

Cover the projective line by the affine t-chart and the affine u-chart with u=1/t. For O(n), take frames e₀ and e∞ with e∞=tⁿe₀ on the overlap. Expressed in e₀, chart sections contribute k[t] and tⁿk[t⁻¹] inside the Laurent ring. Their intersection gives global sections, while the quotient of the Laurent ring by their sum gives first cohomology. The two-chart ordered complex has no higher terms.

When n is nonnegative, the intersection has basis 1,t,…,tⁿ, and every Laurent exponent lies in the sum, so first cohomology vanishes. When n is at most minus two, no exponent is in both chart ranges, but the exponents strictly between n and zero are missing from their sum. They give a basis of first cohomology. Thus dimensions can be computed without mistaking global sections for all functions on a single affine chart; the transition power controls the obstruction.