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

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 $R\to A$ 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 $A{\otimes }_{R}F$ 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.

LaTeX 2e, 13 pages; v.2: misprints corrected, Example 3.10 added at the end; v.3: three references added in the Introduction, Remark 3.3(4) added, a short paragraph inserted near the end of Example 3.10, the proof of Lemma 2.4 and the first paragraph of Remark 3.8 slightly shortened; v.4: two misprints corrected

References in corpus (1)

• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity