Flat cotorsion modules over Noether algebras

TL;DR

This paper characterizes flat cotorsion modules over Noether algebras using prime ideals, extending Enochs' work, and establishes a duality correspondence between indecomposable injective and flat cotorsion modules.

Contribution

It provides a complete description of flat cotorsion modules over Noether algebras and links duality theories with module classifications.

Findings

Characterization of flat cotorsion modules via prime ideals.

Establishment of a bijective correspondence between indecomposable injective and flat cotorsion modules.

Connection of duality theories with module spectrum classifications.

Abstract

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality of Ziegler spectra.