# Some algebraic aspects of mesoprimary decomposition

## TL;DR

This paper refines the algebraic understanding of mesoprimary decomposition of binomial ideals under positive grading, simplifying the theory and providing counterexamples to open questions.

## Contribution

It presents an algebraic approach to mesoprimary decomposition under positive grading, simplifying existing combinatorial methods and addressing open questions with counterexamples.

## Key findings

- Simplified algebraic definitions for mesoprimary decomposition.
- Counterexample showing the hull of a binomial ideal need not be binomial.
- Counterexample for binomiality of toral components in 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. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive $A$-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the $A$-grading), we are able to obtain further simplifications under additional assumptions. We also provide counterexamples to two open questions, identifying (i) a binomial ideal whose hull is not binomial, answering a question of Eisenbud and Sturmfels, and (ii) a binomial ideal $I$ for which $I_\text{toral}$ is not binomial, answering a question of Dickenstein, Miller and the first author.