nLab
coherent module

Contents

Definition

Suppose we are given a (not neccesarily commutative) unital ring RR. A left RR-module MM is finitely generated if there is an exact sequence R nM0R^n\to M\to 0 of left RR-modules where nn is a natural number. MM is a noetherian RR-module if each RR-submodule NMN\subset M is finitely generated. A ring is noetherian if it is noetherian as a left RR-module.

A left RR-module MM is finitely presented (or of finite presentation) if there exists an exact sequence R qR pM0R^q\to R^p\to M\to 0 where p,qp,q are natural numbers. A left coherent module is a left RR-module which is finitely generated and such that every finitely generated RR-submodule NMN\subset M is finitely presented (equivalently: such that the kernel of any (not neccessarily surjective) linear map R nMR^n \to M is finitely generated).

Coherent modules behave well over noetherian rings and to some extent generally over coherent rings.

A geometric globalization of a notion of coherent module is a notion of a coherent sheaf of 𝒪\mathcal{O}-modules for a ringed space (X,𝒪)(X,\mathcal{O}).

References

Revised on September 19, 2016 16:02:37 by Ingo Blechschmidt (95.91.249.167)