Coherent Sheaf - Definition

Definition

A coherent sheaf on a ringed space is a sheaf of -modules with the following two properties:

1. is of finite type over, i.e., for any point there is an open neighbourhood such that the restriction of to is generated by a finite number of sections (in other words, there is a surjective morphism for some ); and
2. for any open set, any and any morphism of -modules, the kernel of is of finite type.

The sheaf of rings is coherent if it is coherent considered as a sheaf of modules over itself. Important examples of coherent sheaves of rings include the sheaf of germs of holomorphic functions on a complex manifold and the structure sheaf of a Noetherian scheme from algebraic geometry.

A coherent sheaf is always a sheaf of finite presentation, or in other words each point has an open neighbourhood such that the restriction of to is isomorphic to the cokernel of a morphism for some integers and . If is coherent, then the converse is true and each sheaf of finite presentation over is coherent.

For a sheaf of rings, a sheaf of -modules is said to be quasi-coherent if it has a local presentation, i.e. if there exist an open cover by of the topological space and an exact sequence

where the first two terms of the sequence are direct sums (possibly infinite) of copies of the structure sheaf.

For an affine variety X with (affine) coordinate ring R, there exists a covariant equivalence of categories between that of quasi-coherent sheaves and sheaf morphisms on the one hand, and R-modules and module homomorphisms on the other hand. In case the ring R is Noetherian, coherent sheaves correspond exactly to finitely generated modules.

Coherence of sheaves is working in the background of some results in commutative algebra, e.g. Nakayama's lemma, which in terms of sheaves says that if is a coherent sheaf, then the fiber if and only if there is a neighborhood of so that .

The role played by coherent sheaves is as a class of sheaves, say on an algebraic variety or complex manifold, that is more general than the locally free sheaf — such as invertible sheaf, or sheaf of sections of a (holomorphic) vector bundle — but still with manageable properties. The generality is desirable, to be able to take kernels and cokernels of morphisms, for example, without moving outside the given class of sheaves.