  In this paper, we obtain two interesting results on homologically smooth
connected cochain DG algebras. More precisely, we show that any Koszul DG
module in $\mathrm{D_{fg}}(A)$ is compact, when $A$ is a homologically smooth
connected cochain DG algebra with a Noetherian cohomology graded algebra
$H(A)$. And we prove that the homologically smoothness of $A$ is equivalent to
$$\mathrm{D_{fg}}(A)=\mathrm{D}^c(A),$$ if $A$ is a Koszul connected cochain DG
algebra such that $H(A)$ is a Noetherian graded algebra with a balanced
dualizing complex.
