A relative version of Bass' theorem about finite-dimensional algebras

TL;DR

This paper extends Bass' theorem to a relative setting, showing that flat modules over certain finite projective ring extensions are summands of filtered modules, linking cotorsion properties over rings.

Contribution

It introduces a relative version of Bass' theorem for finite projective ring homomorphisms, connecting flat and cotorsion modules across rings.

Findings

Every flat left A-module is a summand of an A-module filtered by modules induced from flat R-modules.

A left A-module is cotorsion if and only if its underlying R-module is cotorsion.

The proof utilizes the cotorsion periodicity theorem.

Abstract

As a special case of Bass' theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this result. Let $R\rightarrow A$ be a homomorphism of associative rings such that $A$ is a finitely generated projective right $R$-module. Then every flat left $A$-module is a direct summand of an $A$-module filtered by $A$-modules $A\otimes_RF$ induced from flat left $R$-modules $F$. In other words, a left $A$-module is cotorsion if and only if its underlying left $R$-module is cotorsion. The proof is based on the cotorsion periodicity theorem.