nLab
coherent module

Definition

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

A left $R$-module $M$ is finitely presented (or of finite presentation) if there exists an exact sequence $R^q\to R^p\to M\to 0$ where $p,q$ are natural numbers. A left coherent module is a left $R$-module which is finitely generated and such that every finitely generated $R$-submodule $N\subset M$ is finitely presented (equivalently: such that the kernel of any (not neccessarily surjective) linear map $R^n \to M$ 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 $\mathcal{O}$-modules for a ringed space $(X,\mathcal{O})$.

Related concepts

• quasicoherent module

References

• James Milne, section 6 of Lectures on Étale Cohomology

• B. Kaup, Coherent D-modules, pp. 109–270 in “Algebraic D-modules”, A. Borel ed., Academic Press.