Powers of graphs & applications to resolutions of powers of monomial ideals

TL;DR

This paper explores the use of geometric cell complexes to describe minimal free resolutions of all powers of certain monomial ideals, providing a complete solution for square-free ideals of projective dimension one.

Contribution

It introduces a combinatorial construction of cubical cell complexes supporting resolutions of powers of graphs for specific monomial ideals.

Findings

Constructed a family of cubical cell complexes for these ideals.

Demonstrated the complexes support resolutions of all powers.

Provided a full answer for the case of projective dimension one.

Abstract

This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-free monomial ideals of projective dimension one, by introducing a combinatorial construction of a family of (cubical) cell complexes whose 1-skeletons are powers of a graph that supports the resolution of the ideal.