Homological properties of homologically smooth connected cochain DGAs

TL;DR

This paper characterizes when a connected cochain DG algebra is homologically smooth, Gorenstein, or Calabi-Yau based on properties of its Ext-algebra, generalizing previous results without requiring Koszul conditions.

Contribution

It extends existing characterizations of homologically smooth, Gorenstein, and Calabi-Yau properties to a broader class of DG algebras without the Koszul assumption.

Findings

Homologically smooth and Gorenstein algebras correspond to Frobenius Ext-algebras.

Calabi-Yau algebras correspond to symmetric Frobenius Ext-algebras.

Generalizes previous results by removing the Koszul hypothesis.

Abstract

Assume that $\mathscr{A}$ is a connected cochain DG algebra. We show that $\mathscr{A}$ is homologically smooth and Gorenstein if and only if its $\mathrm{Ext}$-algebra $H(R\Hom_{\mathscr{A}}(\mathbbm{k},\mathbbm{k}))$ is a Frobenius graded algebra. Moreover, $\mathscr{A}$ is Calabi-Yau if and only if the $\mathrm{Ext}$-algebra $H(R\Hom_{\mathscr{A}}(\mathbbm{k},\mathbbm{k}))$ is a symmetric Frobenius graded algebra. These generalize the corresponding results in \cite{HW1} and \cite{HM}, where the additional Koszul hypothesis is needed.