Homologically smooth connected cochain DGAs

Abstract

Let $\mathcal{A}$ be a connected cochain DG algebra such that $H(\mathcal{A})$ is a Noetherian graded algebra. We give some criteria for $\mathcal{A}$ to be homologically smooth in terms of the singularity category, the cone length of the canonical module $k$ and the global dimension of $\mathcal{A}$. For any cohomologically finite DG $\mathcal{A}$-module $M$, we show that it is compact when $\mathcal{A}$ is homologically smooth. If $\mathcal{A}$ is in addition Gorenstein, we get $$\mathrm{CMreg}M={\mathrm{depth}}_{\mathcal{A}}\mathcal{A}+Ext.regM<\infty ,$$ where $\mathrm{CMreg}M$ is the Castelnuovo-Mumford regularity of $M$, ${\mathrm{depth}}_{\mathcal{A}}\mathcal{A}$ is the depth of $\mathcal{A}$ and $Ext.regM$ is the Ext-regularity of $M$.

arXiv admin note: substantial text overlap with arXiv:1301.4382