A family of simplicial resolutions which are DG-algebras

TL;DR

This paper introduces pivot resolutions, a new family of DG-algebra resolutions for monomial ideals that are shorter than Taylor resolutions and provides explicit formulas for their construction over complete intersections.

Contribution

The paper defines pivot resolutions, proves their minimality conditions, and extends their application to explicit Eisenbud-Shamash formulas for monomial ideals over complete intersections.

Findings

Pivot resolutions are always shorter than Taylor resolutions unless already minimal.

The paper characterizes when pivot resolutions are minimal.

Explicit formulas for Eisenbud-Shamash resolutions are derived using pivot resolutions.

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.