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Ideals of Herzog-Northcott type
Planas Vilanova, Francesc d'Assís; O'Carroll, Liam
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
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.
Àrees temàtiques de la UPC::Matemàtiques i estadística
Commutative rings
Herzog ideal
Northcott ideal
almost complete intersection
associative law of multiplicities
Anells commutatius
Àlgebra commutativa
Classificació AMS::13 Commutative rings and algebras::13A General commutative ring theory
Classificació AMS::13 Commutative rings and algebras::13C Theory of modules and ideals
Classificació AMS::13 Commutative rings and algebras::13H Local rings and semilocal rings
Classificació AMS::13 Commutative rings and algebras::13D Homological methods
Attribution-NonCommercial-NoDerivs 3.0 Spain

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