Expansions of monomial ideals and multigraded modules

TL;DR

This paper introduces an expansion functor for multigraded modules that generalizes variable substitution in monomial ideals, revealing new homological properties and applications in combinatorics.

Contribution

It defines and studies the expansion functor, a novel tool for analyzing monomial ideals and multigraded modules, with implications for combinatorial algebra.

Findings

Expansion functor preserves certain homological properties.

Substituting variables by monomial prime ideals generalizes existing operations.

Applications in combinatorial contexts are demonstrated.

Abstract

We introduce an exact functor defined on multigraded modules which we call the expansion functor and study its homological properties. The expansion functor applied to a monomial ideal amounts to substitute the variables by monomial prime ideals and to apply this substitution to the generators of the ideal. This operation naturally occurs in various combinatorial contexts.