A note on homologically smooth connected cochain DG algebras

TL;DR

This paper investigates properties of homologically smooth connected cochain DG algebras, establishing conditions under which Koszul DG modules are compact and characterizing smoothness via derived categories.

Contribution

It proves that Koszul DG modules are compact in certain smooth DG algebras and characterizes smoothness through derived category equivalences.

Findings

Koszul DG modules are compact in homologically smooth algebras with Noetherian cohomology.

Homological smoothness is equivalent to derived category equality under certain conditions.

Provides criteria linking smoothness and derived categories for Koszul 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.