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Let $A$ be a DG-algebra over a field (say $k$). A DG-module $M$ over $A$ is said to be semi-free if it admits an exhaustive filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_p = M$ such that the cones of $M_i \rightarrow M_{i+1}$ are finite direct sums of shifts of finite dimensional free $A$-modules (I insist on the finite dimensional hypothesis which I need for my problem).

In case $A$ is proper and $M$ is finite dimensional (perfect?) DG module over $A$ then the bar-resolution of $M$ is an (infinite) semi-free resolution of $M$.

I am wondering if there is a way to detect that $M$ itself is semi-free from its bar resolution? I am not sure what to expect. Perhaps that one can extract a finite resolution from the bar-resolution when $M$ is semi-free?

Since I am looking for positive result, extra-hypotheses can be made on $A$. I just don't want to assume that $A$ is smooth (since in my example it is not).

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  • $\begingroup$ So, that "finite dimensional" hypothesis you need, what does it mean? $\endgroup$
    – Fernando Muro
    Commented Aug 1, 2020 at 15:01
  • $\begingroup$ finite dimensional free-modules. I have the feeling some people allow infinite dimensional free modules in the construction of the Bar-resolution. $\endgroup$
    – Libli
    Commented Aug 1, 2020 at 19:55
  • $\begingroup$ Finite rank, you probably mean. Your question is not clear because that seems to be your hypothesis on $M$. $\endgroup$
    – Fernando Muro
    Commented Aug 1, 2020 at 20:49
  • $\begingroup$ sure, that is also my hypothesis on $M$. $\endgroup$
    – Libli
    Commented Aug 2, 2020 at 15:32
  • $\begingroup$ then it holds by hypothesis, you don't need the bar resolution. $\endgroup$
    – Fernando Muro
    Commented Aug 2, 2020 at 16:03

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