Hello,


this question is related to [Differential graded structures on free resolution?](https://mathoverflow.net/questions/40282/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



Multiplicative Structures On Free Resolutions
You can always find a free resolution $F$, of $M$ over $S$, such that $F$ is a dg-module over the Koszul complex. It may not be minimal, but in many cases that's not an issue. This is the path taken by Avramov and Buchweitz in their paper "Homological algebra modulo a regular sequence with special attention to codimension 2". In particular these resolutions can be taken functorially.


Also see Avramov's book "Infinite free resolutions" whose main themes are how to put dg-algebra and dg-module structures on free resolutions, and how this helps to study these resolutions.


More recently Dyckerhoff and Polishchuk/Vaintrob have used dg-categorical methods to study the situation you're interested in. See their recent papers on the Arxiv.


