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!


  • 1
    $\begingroup$ Whenever f is not a zero divisor, your "free nonassociative" resolution is indeed weakly equivalent to S/f, and freeness allows it to automatically lift to the endomorphism algebra of a free resolution of any S/f-module. So far as functoriality, whether you use the Koszul algebra or the free nonassociative algebra there are functorial resolutions of M that you want - the obstructions only come if you're trying to work with a specific resolution such as a minimal resolution. In fact, S/f, the Koszul algebra, and the nonassoc. alg. all have equivalent derived categories (Quillen equivalence!). $\endgroup$ Nov 29, 2010 at 16:32
  • $\begingroup$ That sounds very intersting, Tyler, thank you! Could you give references? $\endgroup$
    – Hanno
    Nov 29, 2010 at 18:27
  • $\begingroup$ Unfortuantely I don't remember references immediately - esp for the equivalence between the Koszul algebra and the nonassociative one, I only know the argument - but am sure someone else probably does. So far as functorial resolutions, a standard technique called the "two-sided bar construction" using the forgetful-free adjoint pair between A-modules and sets (or A-modules and some base algebra over which S and S/f are flat) works. And model categories/Quillen equivalences have quite a number of references now. If there are specifics please let me know and I'll try to track something down. $\endgroup$ Nov 29, 2010 at 19:14
  • $\begingroup$ Ok, I will ask more specifically, hopefully it's not too basic: First, what do you mean by the "free nonassociative resolution"? The dg-algebra I thought of is the free associative algebra with generators and relations given above, but this doesn't seem to be what you are talking about. Secondly, what's the precise relation to the general two-sided bar construction you mentioned, und what are good references for the latter? $\endgroup$
    – Hanno
    Nov 30, 2010 at 0:17
  • $\begingroup$ For some blasted reason I have written the word "nonassociative" absolutely everywhere, when I instead mean "associative (noncommutative)" - and obliviously have not figured out why until now. Sorry. $\endgroup$ Dec 4, 2010 at 13:43

1 Answer 1


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.

  • $\begingroup$ Dear Jesse, thank you. Avramovs book is beautiful! I learned from it that it is always possible to construct some resolution with multiplicative structure; however, to me, this seems somehow unnatural, as it is a structure that only particular resolutions possess, while the family of higher homotopies can be put on any resolution. Also thank you for the other references! I thought I knew them, but maybe I should have another look! :-) $\endgroup$
    – Hanno
    Nov 29, 2010 at 18:35
  • $\begingroup$ Dear Hanno, you're right, an algebra structure can always be put on a resolution, but the module structure I'm suggesting is somehow "less costly." By which I mean the resolution is more likely to be minimal. As to the construction not being natural, it really depends on what your goals are. For instance, often at the end of the day one is interested in derived functors, such as \Ext, and in this case it's enough to find one certain resolution with a given structure. $\endgroup$ Nov 29, 2010 at 19:13
  • $\begingroup$ Let me advocate that it's not "unnatural", really, if you think in terms of you asking for a free resolution over a larger algebra than S, like the Koszul algebra or the nonassociative algebra you mentioned. $\endgroup$ Nov 29, 2010 at 19:15

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