Integrally Closed Projectively Equivalent Ideals

McAdam, Stephen; RatliffJr., L. J.; Sally, Judith D.

doi:10.1007/978-1-4612-3660-3_21

Cited by 18 publications

(27 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let I be a regular ideal of the Noetherian ring R. In [14], McAdam, Ratliff, and Sally prove that the set P(I ) of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and eventually periodic. (P(I ) is eventually periodic means there exist I 1 , .…”

Section: Numerical Semigroups and Projectively Full Idealsmentioning

confidence: 98%

“…They also prove [14,Proposition 2.10] that if an ideal J is projectively equivalent to I , then I and J have the same Rees valuations and the values of I and J with respect to these Rees valuations are proportional. Our goal in the present paper is to build on the work in [14] and further develop the relationship between projective equivalence of ideals and Rees valuations.…”

Section: Introductionmentioning

confidence: 94%

“…A number of properties of projective equivalence can be found in [7,8,[11][12][13][14]20,21]. In this section we explore the relation between projectively equivalent ideals and Rees valuation rings.…”

Section: Rees Valuation Rings and Projectively Equivalent Idealsmentioning

confidence: 99%

“…Making use of interesting work of Rees in [23], McAdam, Ratliff, and Sally in [14,Corollary 2.4] prove that the set of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and eventually periodic. They also prove [14,Proposition 2.10] that if an ideal J is projectively equivalent to I , then I and J have the same Rees valuations and the values of I and J with respect to these Rees valuations are proportional. Our goal in the present paper is to build on the work in [14] and further develop the relationship between projective equivalence of ideals and Rees valuations.…”

Section: Introductionmentioning

confidence: 98%

“…The concept of projective equivalence of ideals and the study of ideals projectively equivalent to I was introduced by Samuel in [24] and further developed by Nagata in [15]. Making use of interesting work of Rees in [23], McAdam, Ratliff, and Sally in [14,Corollary 2.4] prove that the set of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and eventually periodic. They also prove [14,Proposition 2.10] that if an ideal J is projectively equivalent to I , then I and J have the same Rees valuations and the values of I and J with respect to these Rees valuations are proportional.…”

Section: Introductionmentioning

confidence: 98%

See 4 more Smart Citations

Projectively equivalent ideals and Rees valuations

et al. 2004

View full text Add to dashboard Cite

Let R be a Noetherian ring. Two ideals I and J in R are projectively equivalent in case the integral closure of I i is equal to the integral closure of J j for some i, j ∈ N + . It is known that if I and J are projectively equivalent, then the set Rees I of Rees valuation rings of I is equal to the set Rees J of Rees valuation rings of J and the values of I and J with respect to these Rees valuation rings are proportional. We observe that the converse also holds. In particular, if the ideal I has only one Rees valuation ring V , then the ideals J projectively equivalent to I are precisely the ideals J such that Rees J = {V }. In certain cases such as: (i) dim R = 1, or (ii) R is a two-dimensional regular local domain, we observe that if I has more than one Rees valuation ring, then there exist ideals J such that Rees I = Rees J , but J is not projectively equivalent to I . If I and J are regular ideals of R, we prove that Rees I ∪ Rees J ⊆ Rees I J with equality holding if dim R 2, but not holding in general if dim R 3. We associate to I and to the set P(I ) of integrally closed ideals projectively equivalent to I a numerical semigroup S(I ) ⊆ N such that S(I ) = N if and only if there exists J ∈ P(I ) for which P(I ) = {(J n ) a | n ∈ N + }.

show abstract

Section: Numerical Semigroups and Projectively Full Idealsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 94%

Section: Rees Valuation Rings and Projectively Equivalent Idealsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

See 3 more Smart Citations

Projectively equivalent ideals and Rees valuations

et al. 2004

View full text Add to dashboard Cite

show abstract

Projectively full ideals and compositions of consistent systems of rank one discrete valuation rings: a survey

Heinzer

Ratliff

Rush

2010

Commutative Algebra

View full text Add to dashboard Cite

Vanishing of differentials along ideals and non-archimedean approximation

Cutkosky

Hübl²

2000

Mathematische Annalen

View full text Add to dashboard Cite

Dedicated to E. Kunz on the occasion of his sixty-fifth birthday One of the major new developments in commutative algebra over the last decade or so was the introduction of the theory of tight closure of an ideal by Hochster and Huneke. It proved to be an extremely useful technique to study ideals, and it also turned out to be closely related to many geometric questions. Fedder [F] used derivations in positive characteristic to obtain characterizations of 2 dimensional graded rational singularities in terms of F-purity and F-injectivity. In an attempt to generalize these techniques and to relate rational singularities with F-rationality, Craig Huneke raised the following problem (c.f [FHH]): Let R be a regular local ring, containing a perfect field k, over which R is essentially of finite type, and let C(R/k) be the subring of derivationally constant elements of R/k (i.e. C(R/k) = {x ∈ R : δ(x) = 0 for all δ ∈ Der k (R)}). Then Huneke asked: (1) If I ⊆ R is an ideal, does there exist a constant l = l(R, I) ∈ N with the following property: If x ∈ R with δ(x) ∈ I n+l then there exists a c ∈ C(R/k) with x − c ∈ I n. (2) If an l as in (1) exists, is it possible to bound it in a way useful for reduction mod p techniques, i.e. if char(k) = 0, does there exist a model R/A, I ⊆ R of R/k, I with A/Z of finite type and a constant l(I) such that l(R/mR, I + m/m) ≤ l(I) for all m ∈ Max(A). A result of the above type has been used successfully by Fedder [F] to relate rationality and F-rationality for two-dimensional graded rings, and a positive answer to the above questions would allow to extend these techniques and results to higher dimension. The

show abstract

Integrally Closed Projectively Equivalent Ideals

Cited by 18 publications

References 5 publications

Projectively equivalent ideals and Rees valuations

Projectively equivalent ideals and Rees valuations

Projectively full ideals and compositions of consistent systems of rank one discrete valuation rings: a survey

Vanishing of differentials along ideals and non-archimedean approximation

Contact Info

Product

Resources

About