# Coherent sheaf

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

on a ringed space $(X,\def\cO{ {\mathcal O}}\cO)$

A sheaf of modules $\def\cF{ {\mathcal F}}\cF$ over a sheaf of rings $\cO$ with the following properties:

1) $\cF$ is a sheaf of finite type, that is, it is locally generated over $\cO$ by a finite number of sections; and

2) the kernel of any homomorphism of sheaves of modules $\cO^p\mid_U\to \cF\mid_U$ over an open set $U\subset X$ is a sheaf of finite type.

If in an exact sequence $0\to \cF_1\to\cF_2\to\cF_3\to 0$ of sheaves of $\cO$-modules two of the three sheaves $\cF_i$ are coherent, then the third is coherent as well. If $\def\phi{\varphi}\phi:\cF\to\def\cS{ {\mathcal S}}\cS$ is a homomorphism of coherent sheaves of $\cO$-modules, then ${\rm Ker}\;\phi$, ${\rm Coker}\;\phi$, ${\rm Im}\;\phi$ are also coherent sheaves. If $\cF$ and $\cS$ are coherent, then so are $\cF\otimes_\cO \cS$ and ${\rm Hom}_\cO(\cF,\cS)$ [Se].

A structure sheaf $\cO$ is called a coherent sheaf of rings if $\cO$ is coherent as a sheaf of modules over itself, which reduces to condition 2). If $\cO$ is a coherent sheaf of rings, then a sheaf of $\cO$-modules $\cF$ is coherent if and only if every point of the space $X$ has a neighbourhood $U$ over which there is an exact sequence of sheaves of $\cO$-modules:

$$\cO^p\mid_U\to\cO^q\mid_U\to\cF\mid_U\to 0,$$ [Se]. Furthermore, under this condition ${\rm Ext}_\cO^p(\cF,\cS)$ is coherent for any coherent sheaves $\cF$, $\cS$ and for all $p$ (see [BaSt]).

The fundamental classes of ringed spaces with a coherent structure sheaf $\cO$ are: analytic spaces over algebraically closed fields [Ab], Noetherian schemes and, in particular, algebraic varieties [Se]. A classical special case is the sheaf $\cO$ of germs of holomorphic functions in a domain of $\mathbf C^n$; the statement that it is coherent is known as the Oka coherence theorem [GuRo], [Fu]. 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=51851
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article