descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 19 Jan 2011Embargo end date: 01 Jan 2010 English Publisher:Cambridge University Press (CUP)Journal:Proceedings of the Edinburgh Mathematical Society, volume 54, pages 161-186 (issn: 0013-0915, eissn: 1464-3839,

Authors: Planas Vilanova, Francesc d'Assís; O'Carroll, Liam;

doi: 10.1017/s0013091509001321 , 10.48550/arxiv.1001.5397

arXiv: 1001.5397

handle: 2117/11327

Ideals of Herzog–Northcott type

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Abstract

AbstractThis 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.

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Universitat Politècnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
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Heriot-Watt University
United Kingdom
Universtity of Edinburgh
United Kingdom
University of Edinburgh
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Keywords

Commutative rings, 13A15, 13C40, 13H15, 13D02, Anells commutatius, liaison, Commutative Algebra (math.AC), Syzygies, resolutions, complexes and commutative rings, Àlgebra commutativa, Herzog ideal, FOS: Mathematics, :13 Commutative rings and algebras::13A General commutative ring theory [Classificació AMS], :13 Commutative rings and algebras::13H Local rings and semilocal rings [Classificació AMS], Classificació AMS::13 Commutative rings and algebras::13A General commutative ring theory, Linkage, complete intersections and determinantal ideals, Àrees temàtiques de la UPC::Matemàtiques i estadística, almost complete intersection, Classificació AMS::13 Commutative rings and algebras::13H Local rings and semilocal rings, :13 Commutative rings and algebras::13C Theory of modules and ideals [Classificació AMS], associative law of multiplicities, Multiplicity theory and related topics, :Matemàtiques i estadística [Àrees temàtiques de la UPC], Northcott ideal, Mathematics - Commutative Algebra, Classificació AMS::13 Commutative rings and algebras::13C Theory of modules and ideals, Classificació AMS::13 Commutative rings and algebras::13D Homological methods, Ideals and multiplicative ideal theory in commutative rings, :13 Commutative rings and algebras::13D Homological methods [Classificació AMS]

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