Ideals of Herzog–Northcott type
| Autor: | Francesc Planas-Vilanova, Liam O'Carroll |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions |
| Rok vydání: | 2011 |
| Předmět: |
Pure mathematics
General Mathematics Prime ideal Commutative rings Splitting of prime ideals in Galois extensions Ideal class group Anells commutatius liaison Àlgebra commutativa Herzog ideal Boolean prime ideal theorem Ascending chain condition on principal ideals 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] Mathematics Discrete mathematics almost complete intersection 13 Commutative rings and algebras::13C Theory of modules and ideals [Classificació AMS] Mathematics::Commutative Algebra associative law of multiplicities Semiprime ring Matemàtiques i estadística [Àrees temàtiques de la UPC] Northcott ideal Fractional ideal Going up and going down 13 Commutative rings and algebras::13D Homological methods [Classificació AMS] |
| Zdroj: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) Recercat. Dipósit de la Recerca de Catalunya instname |
| ISSN: | 1464-3839 0013-0915 |
| DOI: | 10.1017/s0013091509001321 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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