# Does a homologically bounded dg A-module admit a “locally finite” semi-free resolution

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).

• 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. – the L Mar 30 at 19:28
• 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. – mathdonkey Mar 30 at 19:31