# Coherent sheaf

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

on a ringed space A sheaf of modules over a sheaf of rings with the following properties: 1) is a sheaf of finite type, that is, it is locally generated over by a finite number of sections; and 2) the kernel of any homomorphism of sheaves of modules over an open set is a sheaf of finite type. If in an exact sequence of sheaves of -modules two of the three sheaves are coherent, then the third is coherent as well. If is a homomorphism of coherent sheaves of -modules, then , , are also coherent sheaves. If and are coherent, then so are and .

A structure sheaf is called a coherent sheaf of rings if is coherent as a sheaf of modules over itself, which reduces to condition 2). If is a coherent sheaf of rings, then a sheaf of -modules is coherent if and only if every point of the space has a neighbourhood over which there is an exact sequence of sheaves of -modules: . Furthermore, under this condition is coherent for any coherent sheaves , and for all (see ).

The fundamental classes of ringed spaces with a coherent structure sheaf are: analytic spaces over algebraically closed fields , Noetherian schemes and, in particular, algebraic varieties . A classical special case is the sheaf of germs of holomorphic functions in a domain of ; the statement that it is coherent is known as the Oka coherence theorem , . The structure sheaf of a real-analytic space is not coherent, in general.

How to Cite This Entry:
Coherent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coherent_sheaf&oldid=16458
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article