  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.
