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Ideals of Herzog–Northcott type

Part of: Local rings and semilocal rings Commutative algebra: Homological methods Theory of modules and ideals General commutative ring theory

Published online by Cambridge University Press:  19 January 2011

Liam O'Carroll  and
Francesc Planas-Vilanova
Liam O'Carroll
Affiliation:
Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, UK (l.o'carroll@ed.ac.uk)
Francesc Planas-Vilanova
Affiliation:
Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Diagonal 647, ETSEIB, 08028 Barcelona, Spain (francesc.planas@upc.edu)
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Abstract

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This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

Keywords

Herzog idealNorthcott idealalmost complete intersectionliaisonassociative law of multiplicities

MSC classification

Secondary: 13A15: Ideals; multiplicative ideal theory 13C40: Linkage, complete intersections and determinantal ideals 13H15: Multiplicity theory and related topics 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 161 - 186
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Bourbaki, N., Algèbre, in Algèbre homologique, Chapter 10 (Masson, Paris, 1980).Google Scholar
2.Bresinsky, H., Monomial space curves in as set-theoretic complete intersections, Proc. Am. Math. Soc. 75 (1979), 23–24.Google Scholar
3.Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambrige Studies in Advanced Mathematics, Volume 39 (Cambridge University Press, 1993).Google Scholar
4.Eagon, J. A. and Northcott, D. G., Ideals defined by matrices and a certain complex associated with them, Proc. R. Soc. Lond. A 269 (1962), 188–204.Google Scholar
5.Eisenbud, D. and Sturmfels, B., Binomial ideals, Duke Math. J. 84 (1996), 1–45.CrossRefGoogle Scholar
6.Fossum, R. M., The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 74 (Springer, 1973).Google Scholar
7.Gastinger, W., Über die Verschwindungsideale monomialer Kurven, PhD thesis, Universit ät Regensburg (1989).Google Scholar
8.Greuel, G.-M., Pfister, G. and Schönemann, H., Singular: a computer algebra system for polynomial computations, University of Kaiserslautern (available at http:\\www.singular.uni-kl.de).Google Scholar
9.Herrmann, M., Moonen, B. and Villamayor, O., Ideals of linear type and some variants, Queen's Papers in Pure and Applied Mathematics, Volume 83 (Queen's University, Kingston, ON, 1989).Google Scholar
10.Herzog, J., Generators and relations of abelian semigroups and semigroup rings, Manuscr. Math. 3 (1970), 175–193.CrossRefGoogle Scholar
11.Huneke, C., Linkage and the Koszul homology of ideals, Am. J. Math. 104 (1982), 1043–1062.CrossRefGoogle Scholar
12.Huneke, C., The theory of d-sequences and powers of ideals, Adv. Math. 46 (1982), 249–279.>CrossRefGoogle Scholar
13.Kaplansky, I., Commutative rings, revised edn (University of Chicago Press, 1974).Google Scholar
14.Kunz, E., Introduction to commutative algebra and algebraic geometry (Birkhäuser, Boston, MA, 1985).Google Scholar
15.Northcott, D. G., A homological investigation of a certain residual ideal, Math. Annalen 150 (1963), 99–110.CrossRefGoogle Scholar
16.O'Carroll, L. and Planas-Vilanova, F., Irreducible affine space curves and the uniform Artin-Rees property on the prime spectrum, J. Alg. 320 (2008), 3339–3344.CrossRefGoogle Scholar
17.Planas-Vilanova, F., Vanishing of the André-Quillen homology module H 2(A,B,G(I)), Commun. Alg. 24 (1996), 2777–2791.CrossRefGoogle Scholar
18.Swanson, I. and Huneke, C., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, Volume 336 (Cambridge University Press, 2006).Google Scholar
19.Ulrich, B., Lectures on linkage and deformation, ICTP Lecture Notes (ICTP Publications, Trieste, 1992).Google Scholar
20.Valla, G., On determinantal ideals that are set-theoretic complete intersections, Compositio Math. 42 (1981), 3–11.Google Scholar
21.Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, Volume 2 (Springer, 1998).Google Scholar
22.Waldi, R., Zur Konstruktion von Weierstrasspunkten mit vorgegebener Halbgruppe, Manuscr. Math. 30(3) (1980), 257–278.CrossRefGoogle Scholar