Linearization of monomial ideals

TL;DR

This paper introduces a linearization process for monomial ideals that produces ideals with linear quotients and resolutions, along with an auxiliary equification construction that relates to homogenization.

Contribution

It presents a novel linearization method for monomial ideals and explores its properties, including Betti numbers and combinatorial aspects, along with an auxiliary equification construction.

Findings

The linearization produces ideals with linear quotients and resolutions.

The equification relates to homogenization and preserves certain properties.

The constructions have implications for Betti numbers and combinatorial interpretations.

Abstract

We introduce a construction, called linearization, that associates to any monomial ideal $I$ an ideal $\mathrm{Lin}(I)$ in a larger polynomial ring. The main feature of this construction is that the new ideal $\mathrm{Lin}(I)$ has linear quotients. In particular, since $\mathrm{Lin}(I)$ is generated in a single degree, it follows that $\mathrm{Lin}(I)$ has a linear resolution. We investigate some properties of this construction, such as its interplay with classical operations on ideals, its Betti numbers, functoriality and combinatorial interpretations. We moreover introduce an auxiliary construction, called equification, that associates to any monomial ideal a new monomial ideal generated in a single degree, in a polynomial ring with one more variable. We study some of the homological and combinatorial properties of the equification, which can be seen as a monomial analogue of the well-known homogenization construction.