Mesoprimary decomposition of binomial submodules

TL;DR

This paper extends the combinatorial mesoprimary decomposition method from binomial ideals to binomial submodules of graded modules, providing a generalized approach that aligns with prior ideal decomposition techniques.

Contribution

It generalizes mesoprimary decomposition from binomial ideals to binomial submodules of graded modules over monoid algebras, broadening the applicability of the method.

Findings

Provides a combinatorial method for primary decomposition of binomial submodules.

Generalizes existing mesoprimary decomposition techniques to modules.

Ensures the method coincides with Kahle and Miller's approach for binomial ideals.

Abstract

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. These mesoprimary decompositions are highly combinatorial in nature, and are designed to parallel standard primary decomposition over Noetherean rings. In this paper, we generalize mesoprimary decomposition from binomial ideals to "binomial submodules" of certain graded modules over the corresponding monoid algebra, analogous to the way primary decomposition of ideals over a Noetherean ring $R$ generalizes to $R$-modules. The result is a combinatorial method of constructing primary decompositions that, when restricting to the special case of binomial ideals, coincides with the method introduced by Kahle and Miller.