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Irreducible decomposition of binomial ideals

Abstract

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

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[Bass62]Bass, H., Injective dimension in Noetherian rings, Trans. Amer. Math. Soc. 102 (1962), 1829.
[ES96]Eisenbud, D. and Sturmfels, B., Binomial ideals, Duke Math. J. 84 (1996), 145.
[KM14]Kahle, T. and Miller, E., Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 12971364.
[Mil02]Miller, E., Cohen–Macaulay quotients of normal semigroup rings via irreducible resolutions, Math. Res. Lett. 9 (2002), 117128.
[MS05]Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227 (Springer, New York, NY, 2005).
[Vas98]Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2 (Springer, Berlin, 1998).
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Irreducible decomposition of binomial ideals

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