Homological properties of homologically smooth connected cochain DGAs

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.