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1.
Summary LetR be a Cohen-Macaulay ring andI an unmixed ideal of heightg which is generically a complete intersection and satisfiesI
(n)=In for alln≥1. Under what conditions will the Rees algebra be Cohen-Macaulay or have good depth? A series of partial answers to this
question is given, relating the Serre condition (S
r
) of the associated graded ring to the depth of the Rees algebra. A useful device in arguments of this nature is the canonical
module of the Rees algebra. By making use of the technique of the fundamental divisor, it is shown that the canonical module
has the expected form: ω
R[It]
≅(t(1−t)
g−2).
The third author was partially supported by the NSF
This article was processed by the author using theLaTex style filecljour1 from Springer-Verlag. 相似文献
2.
Anurag K. Singh 《Mathematische Zeitschrift》2000,235(2):335-344
Let I be a divisorial ideal of a strongly F-regular ring A. K.-i. Watanabe raised the question whether the symbolic Rees algebra is Cohen-Macaulay whenever it is Noetherian. We develop the notion of multi-symbolic Rees algebras and use this to show that
is indeed Cohen-Macaulay whenever a certain auxiliary ring is finitely generated over A.
Received August 10, 1998 / in final form October 18, 1999 / Published online July 20, 2000 相似文献
3.
《代数通讯》2013,41(8):3713-3734
Abstract Let (R, 𝔪) be a Noetherian local ring and let Ibe an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ? = ?/𝔪? of I, where ? denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous generating relations in low degrees. In particular, if Iis a strongly Cohen-Macaulay R-ideal with G ?and the expected reduction number, we conclude that ? is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ?/K? for any 𝔪-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K = I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ? and the Cohen-Macaulay property of the Rees algebra ? and the associated graded ring 𝒢 of I. Finally, we focus on the integral closedness of 𝔪I. The latter question is motivated by the theory of evolutions. 相似文献
4.
Eero Hyry 《Compositio Mathematica》2002,130(3):319-343
Let A be a local ring, and let I
1,...,I
r
A be ideals of positive height. In this article we compare the Cohen–Macaulay property of the multi–Rees algebra R
A
(I
1,...,I
r
) to that of the usual Rees algebra R
A
(I
1 ··· I
r
) of the product I
1 ··· I
r
. In particular, when the analytic spread of I
1 ··· I
r
is small, this leads to necessary and sufficient conditions for the Cohen–Macaulayness of R
A
(I
1,...,I
r
). We apply our results to the theory of joint reductions and mixed multiplicities. 相似文献
5.
Takesi Kawasaki 《代数通讯》2013,41(12):4385-4396
Let A be a Noetherian ring.We consider the existence of Cohen-Macaulay Rees algebras of A. If the non-Cohen-Macaulay locus of A is of dimension 0, then we already know that such a Rees algebra exists. In the present paper, we show that such a Rees algebra also exists when the non-Cohen-Macaulay locus of A is of dimension 1. 相似文献
6.
Eero Hyry 《manuscripta mathematica》1999,98(3):377-390
Let A be a normal local ring which is essentially finite type over a field of characteristic zero. Let I⊂A be an ideal such that the Rees algebra R
A
(I) is Cohen–Macaulay and normal. In this paper we address the question: “When does R
A
(I) have rational singularities?” In particular, we study the connection between rational singularities of R
A
(I) and the adjoint ideals of the powers I
n
(n∋ℕ).
Received: 25 May 1998 / Revised version: 20 August 1998 相似文献
7.
Duong Quôc Viê.t 《Journal of Pure and Applied Algebra》2006,205(3):498-509
Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals. 相似文献
8.
LetR be a commutative ring,I an invertibleR-module, and consider quadratic spaces with values inI. The Clifford algebra of such a quadratic space is an algebra over the generalized Rees ring associated toI. We discuss the relation between the Witt module of quadratic spaces with values inI and the graded Witt ring and the graded Brauer-Wall group of the generalized Rees ring. This leads to the introduction of three distinguished subgroups of the graded Brauer-Wall group of the generalized Rees ring. The image of the Clifford functor is a subgroup of one of these three subgroups (the type 1 subgroup). 相似文献
9.
William Chin 《Israel Journal of Mathematics》1987,60(2):236-256
In this paper restricted differential operator rings are studied. A restricted differential operator ring is an extension
of ak-algebraR by the restricted enveloping algebra of a restricted Lie algebra g which acts onR. This is an example of a smash productR #H whereH=u (g). We actually deal with a more general twisted construction denoted byR * g where the restricted Lie algebra g is not necessarily embedded isomorphically inR * g. Assume that g is finite dimensional abelian. The principal result obtained is Incomparability, which states that prime
idealsP
1 ⊆P
2 ⊂R * g have different intersections withR. We also study minimal prime ideals ofR * g whenR is g-prime, showing that the minimal primes are precisely those having trivial intersection withR, that these primes are finite in number, and their intersection is a nilpotent ideal. Prime and primitive ranks are considered
as an application of the foregoing results. 相似文献
10.
The core of an R-ideal I is the intersection of all reductions of I. This object was introduced by D. Rees and J. Sally and later studied by C. Huneke and I. Swanson, who showed in particular
its connection to J. Lipman's notion of adjoint of an ideal. Being an a priori infinite intersection of ideals, the core is
difficult to describe explicitly. We prove in a broad setting that: core(I) is a finite intersection of minimal reductions; core(I) is a finite intersection of general minimal reductions; core(I) is the contraction to R of a ‘universal’ ideal; core(I) behaves well under flat extensions. The proofs are based on general multiplicity estimates for certain modules.
Received: 16 May 2000 / Revised version: 11 December 2000 / Published online: 17 August 2001 相似文献
11.
Hà Huy Tài 《manuscripta mathematica》2002,107(4):479-501
Suppose ? is a set of arbitrary number of smooth points in ℙ2
its defining ideal. In this paper, we study the Rees algebras of the ideals generated by I
t
, t ≥α. When the points of ? are general, we give a set of defining equations for the Rees algebra . When the points of ? are arbitrary, we show that for all t≫ 0, the Rees algebra is Cohen-Macaulay and its defining ideal is generated by quadratics. A cohomological characterization for arithmetic Cohen-Macaulayness
of subvarieties of a product space is also given.
Received 4 April 2001 相似文献
12.
Ze Min Zeng 《代数通讯》2013,41(9):3459-3466
Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I 2 is free A[T]/I-module of rank n and (A[T]/I) is torsion in K 0(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers. 相似文献
13.
M. K. Kouakou 《代数通讯》2013,41(1):81-95
Let A 1: = 𝕜[t, ?] be the first algebra over a field 𝕜 of characteristic zero. Let Aut𝕜(A 1) be the automorphism group of the ring A 1. One can associate to each right ideal I of A 1 a subgroup of Aut𝕜(A 1) called the isomorphism subgroup of I. In this article, we show that each such isomorphism subgroup is equal to its normalizer. For that, we study when the isomorphism subgroup of a right ideal of A 1 contains a given isomorphism subgroup. 相似文献
14.
Wolmer V. Vasconcelos> 《Compositio Mathematica》2003,139(3):361-379
Let (Rmbe a Cohen–Macaulay local ring and let I be an ideal. There are at least five algebras built on I whose multiplicity data affect the reduction number r(I) of the ideal. We introduce techniques from the Rees algebra theory of modules to produce estimates for r(I), for classes of ideals of dimension one and two. Previous cases of such estimates were derived for ideals of dimension zero. 相似文献
15.
Let R be a Noetherian ring and let I be an ideal of R. We study when the Rees algebra of I satisfies the condition (S2) of Serre and, when this property is missing, to enable it in a finite extension of R[It]. 相似文献
16.
Guram Donadze 《代数通讯》2013,41(11):4447-4460
We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC *(ρ): HC *(I) → HC *(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence. 相似文献
17.
Francesc Planas-Vilanova 《manuscripta mathematica》1995,87(1):349-357
For a given idealI of a commutative ringA, B=A/I, the vanishing of the second André-Quillen (co)homology functorH
2
(A, B, δ) is characterized in terms of the canonical homomorphism α:S(I)→R(I) from the symmetric algebra of the idealI onto its Rees algebra. This is done by introducing a Koszul complex that characterizes commutative graded algebras which
are symmetric algebras.
This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag. 相似文献
This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag. 相似文献
18.
J.K. Verma 《代数通讯》2013,41(12):2999-3024
Let (R,m) be a local ring. Let SM denote the Rees algebra S=R[mrt] localized at its unique maximal homogeneous ideal M=(m,mrt). Let TN denote the extended Rees algebra T= R[mrt, t-1] localized at its unique maximal homogeneous idea N= (t?1,m,mr). Multiplicity formulas are developedfor SM and TN. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that SM and TN are Cohen-Macaulay with minimal multiplicity 相似文献
19.
We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I>, which consists of all elements of A such that v(a)>v(I), for all Rees valuations v of I. The ideal I> plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition Af and is contained in every reduction of I. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion. 相似文献
20.
Let S=K[x1,…,xn] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring. 相似文献