= Cohomology and base change for line bundles on a curve
In a proper flat family of <algebraic curves>, if the first <sheaf cohomology> of a <line bundle> vanishes on every fibre of an open set, its direct image there is a <vector bundle> of rank given by the <Riemann-Roch theorem> and commutes with arbitrary <base change>. Locally a finite free complex computes both fibre <sheaf cohomology> groups. Vanishing of the last cokernel makes its last map surjective; splitting that map leaves a finite locally free kernel in degree zero. This is the mechanism behind charts of the <Picard scheme of a curve> and evaluation descriptions of the <theta divisor>.
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