Homologically Smooth Connected Cochain DGAs

Let \(\mathscr {A}\) be a connected cochain DG algebra such that \(H(\mathscr {A})\) is a Noetherian graded algebra. We give some criteria for \(\mathscr {A}\) to be homologically smooth in terms of the singularity category, the cone length of the canonical module k and the global dimension of \(\mathscr {A}\). For any cohomologically finite DG \(\mathscr {A}\)-module M, we show that it is compact when \(\mathscr {A}\) is homologically smooth. If \(\mathscr {A}\) is in addition Gorenstein, we get

$$\begin{aligned} \textrm{CMreg}M = \textrm{depth}_{\mathscr {A}}\mathscr {A} + \mathrm {Ext.reg}\, M<\infty , \end{aligned}$$

where \(\textrm{CMreg}M\) is the Castelnuovo-Mumford regularity of M, \(\textrm{depth}_{\mathscr {A}}\mathscr {A}\) is the depth of \(\mathscr {A}\) and \( \mathrm {Ext.reg}\, M\) is the Ext-regularity of M.