A family of simplicial resolutions which are DG-algebras
James Cameron, Trung Chau, Sarasij Maitra, Tim Tribone
TL;DR
This work introduces pivot resolutions as a versatile class of free resolutions for monomial ideals that simultaneously support a DG-algebra structure and are typically shorter than the Taylor resolution. Pivot complexes $\mathcal{T}_{i_1,\dots,i_l}$ are canonically embedded between Lyubeznik and Taylor resolutions, and a precise gap-based criterion (involving a Graded lcm condition) determines when they are true resolutions; the Scarf-number provides a way to identify the smallest pivot resolution and yields Betti-number bounds. The authors prove that pivot resolutions inherit a DG-algebra structure from the Taylor resolution and extend the Eisenbud–Shamash construction by supplying explicit higher homotopies for pivot resolutions, enabling lifting to resolutions over $R=Q/\mathfrak{a}$ for regular sequences $\mathfrak{a}\subseteq I$. Overall, the paper offers a concrete, combinatorial framework for obtaining efficient, multiplicative free resolutions of monomial ideals with explicit homotopy data, enriching the landscape of DG-resolution techniques with practical bounds and constructions.
Abstract
Each monomial ideal over a polynomial ring admits a free resolution which has the structure of a DG-algebra, namely, the Taylor resolution. A pivot resolution of a monomial ideal, which we introduce, is a resolution that is always shorter than the Taylor resolution (unless the Taylor resolution is as short as possible) but still retains a DG-algebra structure. We study the basic properties of this family of resolutions including a characterization of when the construction is minimal. Following the work of Sobieska, we use the explicit nature of pivot resolutions to give formulae for the Eisenbud-Shamash construction of a free resolution of a given monomial ideal over complete intersections.