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:
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.
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Coherent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coherent_sheaf&oldid=16458