相关论文: On semicontinuity of multiplicities in families

We prove that the Hilbert-Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild…

This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and…

We study two important numerical invariants, Hilbert--Kunz multiplicity and $F$-signature, on the spectrum of a Noetherian $\mathbf{F}_p$-algebra $R$ that is not necessarily $F$-finite. When $R$ is excellent, we show that the limits…

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to…

We present a unified approach to the study of Hilbert-Kunz multiplicity, F-signature, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that…

In this paper, we initiate a systematic study of the generalized Hilbert-Kunz multiplicity for families of ideals in a Noetherian local ring (R,m) of positive characteristic, and introduce a new asymptotic invariant called the Amao-type…

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal…

Let F be a finite field of characteristic 2 and h be the element x^3+y^3+xyz of F[[x,y,z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x^q,y^q,z^q,h^j) for q a power of 2. We also…

We define a (perfectoid) mixed characteristic version of $F$-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze and Faltings' normalized length (also developed in the work of Gabber-Ramero).…

Let $(R, \mathfrak{m})$ be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an $\mathfrak{m}$-primary…

We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic.…

Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and…

This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call $p$-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g.,…

We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value…

Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this…

We establish the continuity of Hilbert-Kunz multiplicity and F-signature as functions from a Cohen-Macaulay local ring $(R,\m,k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\m$-adic…

We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist…

We explore the equimultiplicity theory of the $F$-invariants Hilbert--Kunz multiplicity, $F$-signature, Frobenius Betti numbers, and Frobenius Euler characteristic over strongly $F$-regular rings. Techniques introduced in this article…

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in (Blanco-Puerto, 2011). With this tool…