A DG $\Gamma$-structure on the Generalized Taylor Resolution

TL;DR

This paper proves that the generalized Taylor resolution admits a DG algebra structure with divided powers, extending previous results and correcting a sign error, with applications to homotopy Lie algebras and monomial ideals.

Contribution

It extends Avramov's DG algebra structure result to the generalized Taylor resolution and corrects a sign error in recent work, broadening understanding of free resolutions.

Findings

Generalized Taylor resolution admits a DG algebra structure with divided powers.

Correction of a sign error in VandeBogert's product formula.

Applications to homotopy Lie algebras and squarefree monomial ideals.

Abstract

Free resolutions of ideals in commutative rings provide valuable insights into the complexity of these ideals. In 1966, Taylor constructed a free resolution for monomial ideals in polynomial rings, which Gemeda later showed admits a differential graded (DG) algebra structure. In 2002, Avramov proved that the Taylor resolution admits a DG algebra structure with divided powers. In 2007, Herzog introduced the generalized Taylor resolution, which is usually smaller than the original Taylor resolution. Recently, in 2023, VandeBogert showed that the generalized Taylor resolution also admits a DG algebra structure. In this paper, we extend Avramov's result by proving that the generalized Taylor resolution admits a DG algebra structure with divided powers. Along the way, we correct a sign error in VandeBogert's product formula. We provide applications concerning homotopy Lie algebras and resolutions of squarefree monomial ideals.