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Let $A$ be a bounded dg-algebra whose underlying algebra is Noetherian and such that $H^*(A)$ is Noetherian. Let $M$ be a cohomologically bounded dg-module over $A$, whose cohomology groups are finitely generated over $H^*(A)$ (one can also assume if it helps that $M$ is finitely generated over $A$).

Question: Does $M$ admit a semi-free resolution such that each each of the filtered pieces is a \emph{finitely generated} free A-module?

Feel free to tweak the hypotheses because I'd be grateful for any result in this vain (with reference or proof).

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  • $\begingroup$ I think what you are looking for might be in Theorem 11.4.40 of the book arxiv.org/abs/1610.09640, but I did not check if these are the exact conditions. $\endgroup$
    – the L
    Commented Mar 30, 2020 at 19:28
  • $\begingroup$ That does look close to what I want but it assumes that A has no cohomology in positive degrees which I definitely don't want. $\endgroup$
    – mathdonkey
    Commented Mar 30, 2020 at 19:31

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