Coherent Sheaf - Coherent Cohomology

The sheaf cohomology theory of coherent sheaves is called coherent cohomology. It is one of the major and most fruitful applications of sheaves, and its results connect quickly with classical theories.

Using a theorem of Schwartz on compact operators in Frechet spaces, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension. This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kähler manifolds. It plays a major role in the proof of the GAGA equivalence analytic <-> algebraic. An algebraic (and much easier) version of this theorem was proved by Serre. Relative versions of this result for a proper morphism were proved by Grothendieck in the algebraic case and by Grauert and Remmert in the analytic case. For example Grothendieck's result concerns the functor Rf_* or push-forward, in sheaf cohomology. (It is the right derived functor of the direct image of a sheaf.) For a proper morphism in the sense of scheme theory, it was shown that this functor sends coherent sheaves to coherent sheaves. The result of Serre is the case of a morphism to a point.

The duality theory in scheme theory that extends Serre duality is called coherent duality (or Grothendieck duality). Under some mild conditions of finiteness, the sheaf of Kähler differentials on an algebraic variety is a coherent sheaf Ω1. When the variety is non-singular its 'top' exterior power acts as the dualising object; and it is locally free (effectively it is the sheaf of sections of the cotangent bundle, when working over the complex numbers, but that is a statement that requires more precision since only holomorphic 1-forms count as sections). The successful extension of the theory beyond this case was a major step.