nLab coherent module

Definition

Consider a (not necessarily commutative) unital ring $R$. Recall that 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, and that it 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 morphism $R^n \to M$ is finitely generated).

Coherent modules behave well over noetherian rings and to some extent more 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

• dualizable module

• coherent D-module

References

• James Milne, section 6 of Lectures on Étale Cohomology

• B. Kaup, Coherent D-modules, chapter II of Armand Borel et al., Algebraic D-modules, Perspectives in Mathematics, Academic Press, 1987 (djvu)