Hello!


In "Homological Algebra on a Complete Intersection", Eisenbud proves the following: 


Let $A$ be a commutative ring, $M$ be an $A$-module and $F^{\ast}\to M$ an $A$-free resolution. Further, assume that $M$ is annihilated by $I := (x\_1,...,x\_n)$, and that $I$ contains a non zero divisor of $A$. Then there exist maps $s\_{\alpha}: F^{\ast}\to F^{\ast}$, indexed by multiindices $\alpha$ of length $n$, of degree $2|\alpha|-1$, with the following properties:


* $s\_0$ is the differential of $F^{\ast}$.
* $s\_j := s\_{(0,0,...,0,1,0,...,0)}$ is a nullhomotopy for the multiplication by $x\_j$.
* For any $\gamma$ with $|\gamma|\geq 2$ we have 
$\sum\limits\_{\alpha+\beta=\gamma} s\_{\alpha} s\_{\beta} = 0$.


Now he asks the following: Assuming $(x\_1,...,x\_n)$ is regular and $F^{\ast}$ is the minimal free resolution of $M$, is is possible to choose the nullhomotopies $s\_j$ in such a way that $s\_j^2=0$ and $s\_i s\_j = -s\_j s\_i$? I.e. can we make $F^{\ast}$ into a differential graded module over the Koszul-Algebra of $(x\_1,...,x\_n)$? 


I'm interested in this question and would like to know if progress has been made to answer it. Is it possible to choose the $s\_i$ as above, and, if not, how can the obstruction be described?


**Edit:**
Is it possible to handle all the ways the $s\_j$ can be constructed? Does it have something to do with the $A\_{\infty}$-stuff? 


Thank you!


Hanno



Differential graded structures on free resolution?
It is true if the projective dimension of $M$ over $A$ is at most $3$, and counter examples exist when the projective dimension is $4$. 


The first counter example was given in Lucho Avramov's [paper](http://www.jstor.org/pss/2374187)
"Obstructions to the existence of a multiplicative structure on minimal resolutions". A simplified example, ($A=k[t\_1,t\_2,t\_3,t\_4], M= A/(t\_1^2, t\_1t\_2, t\_2t\_3, t\_3t\_4, t\_4^2)$, here you can choose the $x\_i$s to be any regular sequence in the annihilator of $M$)
 together with discussion of related and more recent results can be found in Section 2 of this [note](http://books.google.com/books?id=C5j4xCl8Q80C&lpg=PA1&ots=F6RUMrczXS&dq=Infinite%20free%20resolutions&pg=PA1#v=onepage&q=Infinite%20free%20resolutions&f=false) (available on Avramov's [website](http://www.math.unl.edu/~lavramov2/papers.html)). 


