Journal of Commutative Algebra

Muhly local domains and Zariski's theory of complete ideals

Raymond Debremaeker

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Abstract

Let $(R,\M)$ be a two-dimensional Muhly local domain, that is, a two-dimensional integrally closed Noetherian local domain with algebraically closed residue field and with the associated graded ring $\text{gr}_{\M}R$ an integrally closed domain. In this paper we show that a number of fundamental results of Zariski's theory of complete ideals in two-dimensional regular local rings are not necessarily valid in $R$. However, if the associated graded ring $\text{gr}_{\M}R$ satisfies an additional assumption as in work of Muhly and Sakuma, then we are able to show that ``any product of contracted ideals is contracted'' holds in $R$ if and only if $R$ has minimal multiplicity.

Article information

Source
J. Commut. Algebra, Volume 5, Number 4 (2013), 507-526.

Dates
First available in Project Euclid: 31 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1391192654

Digital Object Identifier
doi:10.1216/JCA-2013-5-4-507

Mathematical Reviews number (MathSciNet)
MR3161744

Zentralblatt MATH identifier
1376.13004

Subjects
Primary: 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Keywords
Regular local ring Muhly local domain complete ideal contracted ideal

Citation

Debremaeker, Raymond. Muhly local domains and Zariski's theory of complete ideals. J. Commut. Algebra 5 (2013), no. 4, 507--526. doi:10.1216/JCA-2013-5-4-507. https://projecteuclid.org/euclid.jca/1391192654


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