Homological properties of homologically smooth connected cochain DGAs

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