Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular

Huneke, Craig; Yao, Yongwei

doi:10.1090/s0002-9939-01-06113-5

Cited by 33 publications

(15 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently Watanabe and Yoshida characterized regularity of a local ring in terms of an asymptotic version of condition (c) in Kunz's Theorem. A simplified proof was given later by Huneke-Yao [25]. This result is analogous to the classical characterization of regularity in terms of the Hilbert-Samuel multiplicity that says that if R is unmixed, then it is regular if and only if e(m, R) = 1.…”

Section: Proofsmentioning

confidence: 77%

The Frobenius endomorphism and homological dimensions

Miller¹

2003

Commutative Algebra

View full text Add to dashboard Cite

Section: Proofsmentioning

confidence: 77%

The Frobenius endomorphism and homological dimensions

Miller¹

2003

Commutative Algebra

View full text Add to dashboard Cite

“…([22, Theorem (2.15)],[14]) Let (A, m, k) be a local ring of characteristic p > 0. Let G = gr m (A) the associated graded ring of A with respect m as above.…”

mentioning

confidence: 99%

Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

Watanabe

Yoshida

2005

Nagoya Mathematical Journal

View full text Add to dashboard Cite

Abstract. In this paper, we investigate the lower bound sHK(p, d) of HilbertKunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on threedimensional local rings. In fact, as a main result, we will prove that sHK(p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurfaceFurthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.

show abstract

“…The Hilbert-Kunz multiplicity of any local ring is at least one. Moreover, under mild hypothesis, it is exactly one if and only if the ring is regular; see Theorem 1.5 of [WY00] and [HY02]. These two facts suggest that Hilbert-Kunz multiplicity is a candidate for a multiplicity theory.…”

Section: Hilbert-kunz Multiplicitymentioning

confidence: 96%

Frobenius-Poincaré function and Hilbert-Kunz multiplicity

Mukhopadhyay¹

2022

Preprint

View full text Add to dashboard Cite

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple (M, R, I) in characteristic p > 0 by proving that for any complex number y, the limit)j e −iyj/p n exists. We prove that the limiting function in the complex variable y is entire and name this function the Frobenius-Poincaré function. We establish various properties of Frobenius-Poincaré functions including its relation with the tight closure of the defining ideal I; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of R I [p n ] as n varies. Our description of Frobenius-Poincaré functions in dimension one and two and other examples raises questions on the structure of Frobenius-Poincaré functions in general.

show abstract

Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular

Abstract: Abstract. We give a new and simple proof that unmixed local rings having Hilbert-Kunz multiplicity equal to 1 must be regular.

Cited by 33 publications

References 5 publications

The Frobenius endomorphism and homological dimensions

The Frobenius endomorphism and homological dimensions

Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

Frobenius-Poincaré function and Hilbert-Kunz multiplicity

Contact Info

Product

Resources

About