On the Dimension of Modules and Algebras, VII. Algebras with Finite-Dimensional Residue-Algebras

Abstract

It was shown in Eilenberg-Nagao-Nakayama [3] (Theorem 8 and § 4) that if Ω is an algebra (with unit element) over a field K with (Ω: K) <∞ and if the cohomolgical dimension of Ω, dim Ω, is ≦ 1, then every residue-algebra of Ω has a finite cohomological dimension. In the present note we prove a theorem of converse type, which gives, when combined with the cited result, a rather complete general picture of algebras whose residue-algebras are all of finite cohomological dimension.