On Constructions Related to the Generalized Taylor Complex

TL;DR

This paper extends the theory of the Taylor complex to a generalized version, introduces a generalized Scarf complex, and explores properties of quasitransverse monomial ideals, enhancing understanding of their algebraic and homological structures.

Contribution

It constructs an explicit DG-algebra structure on the generalized Taylor complex, generalizes the Scarf complex, and studies properties of quasitransverse monomial ideals.

Findings

Generalized Taylor complex admits a DG-algebra structure.

Generalized Scarf complex is a direct summand of minimal free resolutions.

Quasitransverse monomial ideals relate to Golodness and Koszul homology.

Abstract

In this paper, we extend constructions and results for the Taylor complex to the generalized Taylor complex constructed by Herzog. We construct an explicit DG-algebra structure on the generalized Taylor complex and extend a result of Katth\"an on quotients of the Taylor complex by DG-ideals. We introduce a generalization of the Scarf complex for families of monomial ideals, and show that this complex is always a direct summand of the minimal free resolution of the sum of these ideals. We also give an example of an ideal where the generalized Scarf complex strictly contains the standard Scarf complex. Moreover, we introduce the notion of quasitransverse monomial ideals, and prove a list of results relating to Golodness, Koszul homology, and other homological properties for such ideals.