Pruning free resolutions over a complete intersection

Abstract

Given a free resolution $F_\bullet$ over a polynomial ring $R$ and a system of higher homotopies on it for a regular sequence $a_1,\dots,a_c$, we can use the Eisenbud-Shamash construction to build a free resolution over $\overline{R}=R/(a_1,\dots,a_c)$. We provide a combinatorial formula for the new system of higher homotopies on a pruned free resolution $F_{\bullet}^{\mathcal{M}}$ that corresponds to a pruning of the Eisenbud-Shamash resolution.