Some algebraic aspects of mesoprimary decomposition Academic Article uri icon

abstract

  • 2018 Elsevier B.V. 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 Itoral is not binomial, answering a question of Dickenstein, Miller and the first author.

published proceedings

  • JOURNAL OF PURE AND APPLIED ALGEBRA

altmetric score

  • 1

author list (cited authors)

  • Matusevich, L. F., & O'Neill, C.

citation count

  • 2

complete list of authors

  • Matusevich, Laura Felicia||O'Neill, Christopher

publication date

  • January 2019