A converse to a construction of Eisenbud–Shamash

Abstract

Let $\left(Q,\mathfrak{𝔫},k\right)$ be a commutative local Noetherian ring, $f_1,\dots ,f_c$ a $Q$-regular sequence in $\mathfrak{𝔫}$, and $R=Q∕\left(f_1,\dots ,f_c\right)$. Given a complex of finitely generated free $R$-modules, we give a construction of a complex of finitely generated free $Q$-modules having the same homology. A key application is when the original complex is an $R$-free resolution of a finitely generated $R$-module. In this case our construction is a sort of converse to a construction of Eisenbud and Shamash which yields a free resolution of an $R$-module $M$ over $R$ given one over $Q$.