Uniform behaviour of the Frobenius closures of ideals generated by regular sequences

Katzman, Mordechai; Sharp, Rodney Y.

doi:10.1016/j.jalgebra.2005.01.025

Cited by 30 publications

(40 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) If R is Cohen-Macaulay then F te(R) = HSL(R) by Katzman and Sharp [8]. In general, the authors of this paper proved in [7] that F te(R) ≥ HSL(R).…”

Section: Preliminariesmentioning

confidence: 93%

See 1 more Smart Citation

Upper bound of multiplicity in prime characteristic

Huong

Quy

2019

Forum Mathematicum

View full text Add to dashboard Cite

show abstract

“…(1) If R is Cohen-Macaulay then F te(R) = HSL(R) by Katzman and Sharp [8]. In general, the authors of this paper proved in [7] that F te(R) ≥ HSL(R).…”

Section: Preliminariesmentioning

confidence: 93%

“…It is asked by Katzman and Sharp that whether F te(R) < ∞ for every (equidimensional) local ring (cf. [8] F te(R) = HSL(R). Moreover the question of Katzman and Sharp has affirmative answers when R is either generalized Cohen-Macaulay by [5] or F -nilpotent by [14] (see the next section for the details).…”

Section: Introductionmentioning

confidence: 99%

Upper bound of multiplicity in prime characteristic

Huong

Quy

2019

Forum Mathematicum

View full text Add to dashboard Cite

show abstract

“…By Proposition 1.8, we have ann E ( n∈N 0 cx n ) = ann E (c), and this is an R[x, f ]-submodule of E that is annihilated by c. Use overlines to denote natural images in R/c of elements of R. It is easy to use [11,Lemma 1.3] to see that ann E (c) inherits a structure as left (R/c)[x, f ]-module with re = re for all r ∈ R and e ∈ ann E (c) (and multiplication by x on an element of ann E (c) as in the R[x, f ]-module E). Note that ann E (c) is x-torsion-free, and that I R/c ann E (c) = d/c: d ∈ I(E) and d ⊇ c .…”

Section: Notesmentioning

confidence: 99%

Graded annihilators and tight closure test ideals

Sharp

2009

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

Let R be a commutative Noetherian local ring of prime characteristic p, with maximal ideal m. The main purposes of this paper are to show that if the injective envelope E of R/m has a structure as an x-torsionfree left module over the Frobenius skew polynomial ring over R (in the indeterminate x), then R has a tight closure test element (for modules) and is F -pure, and to relate the test ideal of R to the smallest 'E-special' ideal of R of positive height.A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where R is an F -pure homomorphic image of an F -finite regular local ring, that there exists a strictly ascending chain 0 = τ 0 ⊂ τ 1 ⊂ · · · ⊂ τ t = R of radical ideals of R such that, for each i = 0, . . . , t − 1, the reduced local ring R/τ i is F -pure and its test ideal (has positive height and) is exactly τ i+1 /τ i . This paper presents an analogous result in the case where R is complete (but not necessarily F -finite) and E has a structure as an x-torsion-free left module over the Frobenius skew polynomial ring. Whereas Cowden Vassilev's results were based on R. Fedder's criterion for F -purity, the arguments in this paper are based on the author's work on graded annihilators of left modules over the Frobenius skew polynomial ring.

show abstract

“…, n l ∈ N + be any positive integers. Then As in [10], use will be made of the following extension, due to G. Lyubeznik, of a result of R. Hartshorne and R. Speiser. It shows that, when R is local and of prime characteristic p, a T -torsion left R[T , f ]-module which is Artinian (that is, 'cofinite' in the terminology of Hartshorne and Speiser) as an R-module exhibits a certain uniformity of behaviour.…”

Section: Corollary 37mentioning

confidence: 99%

Frobenius test exponents for parameter ideals in generalized Cohen–Macaulay local rings

Huneke¹,

Katzman²,

Sharp³

et al. 2006

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal a of R, there is a power Q of p, depending on a, such that the Qth Frobenius power of the Frobenius closure of a is equal to the Qth Frobenius power of a. The paper addresses the question as to whether there exists a uniform Q 0 which 'works' in this context for all parameter ideals of R simultaneously.In a recent paper, Katzman and Sharp proved that there does exists such a uniform Q 0 when R is CohenMacaulay. The purpose of this paper is to show that such a uniform Q 0 exists when R is a generalized Cohen-Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong d-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne-Speiser-Lyubeznik Theorem employed by Katzman and Sharp in the Cohen-Macaulay case. (C. Huneke), m.katzman@sheffield.ac.uk (M. Katzman), r.y.sharp@sheffield.ac.uk (R.Y. Sharp), ywyao@umich.edu (Y. Yao).

show abstract

Uniform behaviour of the Frobenius closures of ideals generated by regular sequences

Cited by 30 publications

References 18 publications

Upper bound of multiplicity in prime characteristic

Upper bound of multiplicity in prime characteristic

Graded annihilators and tight closure test ideals

Frobenius test exponents for parameter ideals in generalized Cohen–Macaulay local rings

Contact Info

Product

Resources

About