Lcm-lattice, Taylor Bases and Minimal Free Resolutions of a Monomial ideal

TL;DR

This paper introduces a novel approach to constructing minimal free resolutions of monomial ideals using their lcm-lattice, including the new concept of a Taylor basis, and offers an approximation method for these resolutions.

Contribution

It develops the atomic lattice resolution theory, linking lcm-lattices to minimal free resolutions, and refines the theory of poset resolutions with an approximation formula.

Findings

Constructed minimal free resolutions from lcm-lattices.

Introduced the Taylor basis concept for resolutions.

Provided an approximation formula for all monomial ideals.

Abstract

We use the lcm-lattice of a monomial ideal to study its minimal free resolutions. A new concept called a Taylor basis of a minimal free resolution is introduced and then used throughout the paper. We give a method of constructing minimal free resolutions of a monomial ideal from its lcm-lattice, which is called the atomic lattice resolution theory. Some applications of this theory is given. As the main application, we rewrite the theory of poset resolutions, and we obtain an approximation formula for minimal free resolutions of all monomial ideals.