Reflecting Perfection for Finite Dimensional Differential Graded Algebras

TL;DR

This paper extends key algebraic properties to finite dimensional differential graded algebras, including Nakayama Lemma and simple modules detecting projective dimension, with dual versions linked to Gorenstein algebras and Koszul duality.

Contribution

It generalizes fundamental algebraic facts to the dg setting and introduces dual versions related to Gorenstein properties and Koszul duality.

Findings

Generalization of Nakayama Lemma to dg algebras

Detection of finite projective dimension by simples in dg context

Corepresentability result for finite dimensional dg algebras

Abstract

We generalise two facts about finite dimensional algebras to finite dimensional differential graded algebras. The first is the Nakayama Lemma and the second is that the simples can detect finite projective dimension. We prove two dual versions which relate to Gorenstein differential graded algebras and Koszul duality respectively. As an application, we prove a corepresentability result for finite dimensional differential graded algebras.