Complete Intersections in Regular Local Rings ∗ (original) (raw)
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Noetherian intersections of regular local rings of dimension two
Journal of Algebra, 2020
Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We examine the Noetherian rings that are intersections of rings in Q(D). To do so, we describe the desingularization of projective models over D both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree Q(D).
On the Gorensteinness of Rees and form rings of almost complete intersections
Nagoya Mathematical Journal
LetAbe a Noetherian local ring andpa prime ideal inA. Letand call them, respectively, the Rees ring and the form ring ofp. The purpose of this paper is to clarify, providedpis an almost complete intersection inA(cf. (2.1) for definition), when the ringsR(p) andG(p) are Gorenstein.
The Rees Valuations of Complete Ideals in a Regular Local Ring
Communications in Algebra, 2015
Let I be a complete m-primary ideal of a regular local ring (R, m) of dimension d ≥ 2. In the case of dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points Lipman proves a unique factorization involving special *-simple complete ideals and possibly negative exponents of the factors. Let T be an infinitely near point to R with dim R = dim T and R/ m = T / mT. We prove that the special *-simple complete ideal PRT has a unique Rees valuation if and only if either dim R = 2 or there is no change of direction in the unique finite sequence of local quadratic transformations from R to T. We also examine conditions for a complete ideal to be projectively full.
Noncomplete intersection prime ideals in dimension 333
Kyoto Journal of Mathematics, 2015
We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.
arXiv (Cornell University), 2023
Let R be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate R. We restrict to the class of prime divisors that dominate R and show that if a collection of such prime divisors is taken below a certain "level," then the intersection is an almost Dedekind domain having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals.
Journal of Algebra and Related Topics, 2021
This paper is a continuation of study rings relative to rightideal, where we study the concepts of regular and local ringsrelative to right ideal. We give some relations between P−P-P−local($P-$regular) and local (regular) rings. New characterizationobtained include necessary and sufficient conditions of a ring RRRto be regular, local ring in terms P−P-P−regular, P−P-P−local ofmatrices ring M2(R)M_{2}(R)M2(R). Also, We proved that every ring is localrelative to any maximal right ideal of it.
“P1-GLUING” for Local Complete Intersections
2017
We prove an analogue of the Affine Horrocks’ Theorem for local complete intersection ideals of height n in R[T ], where R is a regular domain of dimension d, which is essentially of finite type over an infinite perfect field of characteristic unequal to 2, and 2n ≥ d+ 3. Dedicated to Professor S. M. Bhatwadekar on his seventieth birthday.
On ideals with the Rees property
Archiv der Mathematik, 2013
A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal J ⊂ S which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal I ⊂ S is said to be m-full if mI : y = I for some y ∈ m, where m is the graded maximal ideal of S. It was proved by one of the authors that m-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not m-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.