Hello,

this question is related to Differential graded structures on free resolution?.

Given a regular local ring $S$ and $f\in{\mathfrak m}_S\setminus\{0\}$, I am interested in studying $R$-modules through their $S$-free resolutions. More precisely, given an $R$-module $M$, any $S$-free resolution $F^{\ast}$ of $M$ admits homotopies $s_n$ of respective degree $2n-1$ such that $s_0$ equals the differential, $s_1$ is a nullhomotopy for the multiplication with $f$ and for the higher $s_n$ we have

$s_0 s_n + s_1 s_{n-1} + ... + s_n s_0 = 0$ for $n\geq 2$.

If $s_1$ can be chosen in such a way that $s_1^2=0$, we can consider $F^{\ast}$ as a dg-module over the Koszul dg-S-algebra of $S/f$.

Now I have two questions:

(1) What happens if the $s_1$ can *not* be chosen such that $s_1^2=0$? Can we still write the datum of higher homotopies $s_n$ as a dg-module structure over some dg-resolution of $S/f$? A naive guess would be a free non-commutative dg-algebra generated by elements $s_n$ of degree $2n-1$ such that $\text{d}(s_n) = s_1 s_{n-1} + ... + s_{n-1} s_1$ for $n\geq 2$ and $\text{d}(s_1) = f\cdot 1$. Is this studied anywhere?

(2) Can we do all this somehow functorially in $M$? I'm thinking of something like the canonical functor from R-mod into the derived category, which after identification with the homotopy category of projectives turns "projective resolution" into a functor. However, in the concrete example I'm struggling with extending a morphism between R-modules to a morphism of free resolutions respecting the chosen homotopies. Perhaps this is just some diagram chase, but I don't see it.

*Apart from technicalities, my goal is to study $S/f$-modules through dg-modules over an appropriate dg-$S$-algebra up to homotopy. If this is possible: do you know books or articles where it is treated?*

Thank you very much!

Hanno