nLab
coherent module

Contents

Contents

Definition

Consider a (not necessarily commutative) unital ring RR. Recall that 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, and that it 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 morphism R nMR^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 𝒪\mathcal{O}-modules for a ringed space (X,𝒪)(X,\mathcal{O}).

References

Last revised on October 2, 2021 at 10:09:25. See the history of this page for a list of all contributions to it.