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

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. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.