Iterated Mapping Cones for Strongly Koszul Algebras

TL;DR

This paper extends the iterated mapping cone technique to monomial ideals in strongly Koszul algebras, providing explicit resolutions and generalizing known results from polynomial rings.

Contribution

It introduces a method to compute minimal free resolutions of monomial ideals in strongly Koszul algebras using iterated mapping cones and regular orderings.

Findings

Minimal free resolution of monomial ideals with linear quotients in strongly Koszul algebras is obtained via iterated mapping cones.

Explicit formulas for comparison maps in the resolution construction are provided.

The approach generalizes classical results from polynomial rings to strongly Koszul algebras.

Abstract

In this paper we extend the well-known iterated mapping cone procedure to monomial ideals in strongly Koszul algebras. We study properties of ideals generated by monomials in commutative Koszul algebras and show that the linear strand of ideals generated by linear forms is obtained as a subcomplex of the Priddy complex. In the case of strongly Koszul algebras, this shows that the minimal free resolution of a monomial ideal admitting linear quotients is obtained as an iterated mapping cone, immediately extending results for such ideals in polynomial rings to strongly Koszul algebras. We then consider monomial ideals admitting a so-called regular ordering, a generalization of regular decomposition functions, and show that the comparison maps in the iterated mapping cone construction can be computed explicitly. In particular, this gives a closed form for the minimal free resolution of monomial ideals admitting a regular ordering over strongly Koszul algebras.