A note on homologically smooth connected cochain DG algebras

Abstract: 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.