Minimal and cellular free resolutions over polynomial OI-algebras

TL;DR

This paper extends the concept of minimal free resolutions to polynomial OI-algebras, establishing existence, uniqueness, and conditions for width-wise minimal resolutions, especially for certain monomial ideals.

Contribution

It introduces and studies minimal and width-wise minimal free resolutions over polynomial OI-algebras, including existence, uniqueness, and specific cases where width-wise minimal resolutions exist.

Findings

Any finitely generated graded module over a noetherian polynomial OI-algebra has a unique minimal free resolution.

Width-wise minimal free resolutions exist for certain monomial OI-ideals like Ferrers and squarefree strongly stable ideals.

Cellular resolutions are used to establish the existence and properties of these resolutions.

Abstract

Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a polynomial OI-algebra, namely minimal and width-wise minimal free resolutions. A minimal free resolution of an OI-module can be characterized by the fact that the free module in every fixed homological degree, say $i$, has minimal rank among all free resolutions of the module. We show that any finitely generated graded module over a noetherian polynomial OI-algebra admits a graded minimal free resolution, and that it is unique. A width-wise minimal free resolution is a free resolution that provides a minimal free resolution of a module in every width. Such a resolution is necessarily minimal. Its existence is not guaranteed. However, we show that certain monomial OI-ideals do admit width-wise minimal free or, more generally, width-wise minimal flat resolutions. These ideals include families of well-known monomial ideals such as Ferrers ideals and squarefree strongly stable ideals. The arguments rely on the theory of cellular resolutions.