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The primary components of positive critical binomial ideals. (English) Zbl 1274.13002
Authors’ abstract: A natural candidate for a generating set of the (necessarily prime) defining ideal of an \(n\)-dimensional monomial curve, when the ideal is an almost complete intersection, is a full set of \(n\) critical binomials. In a somewhat modified and more tractable context, we prove that, when the exponents are all positive, critical binomial ideals in our sense are not even unmixed for \(n \geq 4\), whereas for \(n \leq 3\) they are unmixed. We further give a complete description of their isolated primary components as the defining ideals of monomial curves with coefficients. This answers an open question on the number of primary components of Herzog-Northcott ideals, which comprise the case \(n=3\). Moreover, we find an explicit, concrete description of the irredundant embedded component (for \(n \geq 4\)) and characterize when the hull of the ideal, i.e., the intersection of its isolated primary components, is prime. Note that these last results are independent of the characteristic of the ground field. Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and Laurent polynomial rings, together with theory of Smith Normal Form and of Fitting ideals. This gives a more transparent and completely general approach, replacing the theory of multiplicities used previously to treat the particular case \(n=3\).

13A05 Divisibility and factorizations in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
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[1] Alcántar, A.; Villarreal, R. H., Critical binomials of monomials curves, Comm. Algebra, 22, 8, 3037-3052, (1994) · Zbl 0855.13014
[2] Bruns, W.; Herzog, J., Cohen-Macaulay rings, Cambridge Stud. Adv. Math., vol. 39, (1993), Cambridge University Press · Zbl 0788.13005
[3] Cox, D.; Little, J.; OʼShea, D., Using algebraic geometry, Grad. Texts in Math., vol. 185, (2005), Springer New York
[4] Dickenstein, A.; Matusevich, L.; Miller, E., Combinatorics of binomial primary decomposition, Math. Z., 264, 4, 745-763, (2010) · Zbl 1190.13017
[5] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math., vol. 150, (1995), Springer-Verlag New York · Zbl 0819.13001
[6] Eisenbud, D.; Sturmfels, B., Binomial ideals, Duke Math. J., 84, 1, 1-45, (1996) · Zbl 0873.13021
[7] Eto, K., Almost complete intersection monomial curves in \(\mathbb{A}^4\), Comm. Algebra, 22, 13, 5325-5342, (1994) · Zbl 0838.14022
[8] Fossum, R. M., The divisor class group of a Krull domain, Ergeb. Math. Grenzgeb., vol. 74, (1973), Springer · Zbl 0256.13001
[9] W. Gastinger, Über die Verschwindungsideale monomialer Kuerven, PhD dissertation, Universität Regensburg, 1989.
[10] Greuel, G.-M.; Pfister, G.; Schönemann, H., \scsingular. A computer algebra system for polynomial computations, University of Kaiserslautern · Zbl 1344.13002
[11] Herrmann, M.; Moonen, B.; Villamayor, O., Ideals of linear type and some variants, (The Curves Seminar at Queenʼs, vol. VI, Kingston, ON, 1989, Queenʼs Papers in Pure and Appl. Math., vol. 83, (1989), Queenʼs Univ. Kingston, ON), Exp. No. H, 37 pp · Zbl 0716.13004
[12] Herzog, J., Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math., 3, 175-193, (1970) · Zbl 0211.33801
[13] Hoşten, S.; Shapiro, J., Primary decomposition of lattice basis ideals, J. Symbolic Comput., 29, 625-639, (2000) · Zbl 0968.13003
[14] Jacobson, N., Basic algebra. I, (1985), W. H. Freeman and Company New York · Zbl 0557.16001
[15] Kahle, T.; Miller, E., Decompositions of commutative monoid congruences and binomial ideals · Zbl 1341.20062
[16] Kaplansky, I., Commutative rings, (1974), The University of Chicago Press Chicago, IL, London · Zbl 0203.34601
[17] Katsabekis, A.; Ojeda, I., An indispensable classification of monomial curves in \(\mathbb{A}_k^4\) · Zbl 1303.13023
[18] López, H. H.; Villarreal, R. H., Computing the degree of a lattice ideal of dimension one · Zbl 1322.13009
[19] Miller, E., Theory and applications of lattice point methods for binomial ideals · Zbl 1251.14041
[20] Miller, E.; Sturmfels, B., Combinatorial commutative algebra, Grad. Texts in Math., vol. 227, (2005), Springer-Verlag New York · Zbl 1090.13001
[21] Northcott, D. G., A homological investigation of a certain residual ideal, Math. Ann., 150, 99-110, (1963) · Zbl 0112.02904
[22] OʼCarroll, L.; Planas-Vilanova, F., Irreducible affine space curves and the uniform Artin-Rees property on the prime spectrum, J. Algebra, 320, 3339-3344, (2008) · Zbl 1152.13013
[23] OʼCarroll, L.; Planas-Vilanova, F., Ideals of herzog-northcott type, Proc. Edinb. Math. Soc. (2), 54, 1, 161-186, (2011) · Zbl 1238.13010
[24] Ojeda, I., Binomial canonical decompositions of binomial ideals, Comm. Algebra, 39, 10, 3722-3735, (2011) · Zbl 1242.13024
[25] Reyes, E.; Villarreal, R. H.; Zárate, L., A note on affine toric varieties, Linear Algebra Appl., 318, 1-3, 173-179, (2000) · Zbl 0986.14032
[26] Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, Algorithms Comput. Math., vol. 2, (1998), Springer-Verlag Berlin
[27] Villarreal, R. H., Monomial algebras, Monogr. Textbooks Pure Appl. Math., vol. 238, (2001), Marcel Dekker, Inc. New York · Zbl 1002.13010
[28] Waldi, R., Zur konstruktion von weierstrasspunkten mit vorgegebener halbgruppe, Manuscripta Math., 30, 257-278, (1980) · Zbl 0473.14007
[29] Zariski, O.; Samuel, P., Commutative algebra, vol. II, Grad. Texts in Math., vol. 29, (1975), Springer-Verlag New York-Heidelberg · Zbl 0121.27901
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