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Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,e_n| \partial e_i=f_i]$ of $\mathbb f$ over $Q$ carries a natural DG-algebra structure. Let $M$ be a finitely generated $Q/I$-module, hence also a $Q$-module.

Let $F_M$ be a minimal $Q$-free resolution of $M$. Note that this resolution is bounded as $Q$ is regular. My question is: Does $F_M$ have a DG $E$-algebra structure? If this is not necessarily true, what if I also assumed $M\cong Q/J$ for some ideal $J$ containing $I$ ?

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This is not true, even if your $Q/I$ is a complete intersection, and so $E$ is the minimal resolution of $Q/I$ over $Q$. See Avramov, “Obstructions to the Existence of Multiplicative Structures on Minimal Free Resolutions.” American Journal of Mathematics, vol. 103, no. 1, 1981, pp. 1–31 for examples where this fails.

What is true is that $F_M$ is an $A_{\infty}$-module over $E$.

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