Sheaf Cohomology

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants.

Its development was rapid in the years after 1950, when it was realised that sheaf cohomology was connected with more classical methods applied to the Riemann-Roch theorem, the analysis of a linear system of divisors in algebraic geometry, several complex variables, and Hodge theory. The dimensions or ranks of sheaf cohomology groups became a fresh source of geometric data, or gave rise to new interpretations of older work.