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Irreducible decomposition of binomial ideals

Published online by Cambridge University Press:  01 April 2016

Thomas Kahle
Affiliation:
Faculty of Mathematics, Otto-von-Guericke Universität, Magdeburg, Germany email thomas.kahle@ovgu.de
Ezra Miller
Affiliation:
Mathematics Department, Duke University, Durham, NC 27708, USA email ezra@math.duke.edu
Christopher O’Neill
Affiliation:
Mathematics Department, Texas A&M University, TX 77843, USA email coneill@math.tamu.edu
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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].

Type
Research Article
Copyright
© The Authors 2016 

References

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