Abstract
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
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.
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The author are grateful to Jiwei He, Xingting Wang and James Zhang for helpful discussions.
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The author was supported by NSFC (Grant No.11871326).
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Mao, XF. Homologically Smooth Connected Cochain DGAs. Algebr Represent Theor (2024). https://doi.org/10.1007/s10468-024-10287-5
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DOI: https://doi.org/10.1007/s10468-024-10287-5