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Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex
Abstract
Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}({\mathfrak {n}}) be the Rees algebra of the maximal homogeneous ideal n {\mathfrak {n}} of R . R. We show that the generalized Hilbert-Kunz function H K ( s ) = ℓ ( R ( n ) / ( n , n t ) [ s ] ) HK(s)=\ell ({\mathcal {R}}({\mathfrak {n}})/({\mathfrak {n}}, {\mathfrak {n}} t)^{[s]}) is given by a polynomial for all large s . s. We calculate it in many examples and also provide a Macaulay2 code for computing H K ( s ) . HK(s).
... Proof. The result follows using the same arguments as in the proof of[1, Corollary 3.3]. ...
We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an -primary ideal exists in a Noetherian local ring with prime characteristic This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring R of a simplicial complex and an ideal J generated by pure powers of the variables, the generalized Hilbert-Kunz function is a polynomial for all q,k and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of J in terms of Hilbert-Samuel multiplicity of J. We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.
We prove that the Hilbert-Kunz function of the ideal (I, It) of the Rees algebra R(I), where I is an m-primary ideal of a 1-dimensional local ring (R,m), is a quasi-polynomial in e, for large e. For s∈N, we calculate the Hilbert-Samuel function of the R-module I[s] and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal (I,It)R(I) when I is a parameter ideal in a Cohen-Macaulay local ring of dimension d≥2, proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case.
Let ( R , ) be an analytically unramified local ring of positive prime characteristic p . For an ideal I , let I * denote its tight closure. We introduce the tight Hilbert function and the corresponding tight Hilbert polynomial , where I is an m-primary ideal. It is proved that F -rationality can be detected by the vanishing of the first coefficient of . We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.
Let I be an ideal in a Noetherian local ring (R, m). It is shown that for n ≫ 0 the reduction number rJ(In) of In with respect to a minimal reduction J is not only independent from the choice of J but also is stable. If I is an m-primary ideal, we give a criterion for the Rees algebra R[ Int ] with n ≫ 0 to be Cohen-Macaulay.
In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese subrings.
We prove that the Hilbert-Kunz function of the ideal (I, It) of the Rees algebra R(I), where I is an m-primary ideal of a 1-dimensional local ring (R,m), is a quasi-polynomial in e, for large e. For s∈N, we calculate the Hilbert-Samuel function of the R-module I[s] and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal (I,It)R(I) when I is a parameter ideal in a Cohen-Macaulay local ring of dimension d≥2, proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case.
This paper is a much expanded version of two talks given in Ann Arbor in May of 2012 during the computational workshop on F-singularities. We survey some of the theory and results concerning the Hilbert-Kunz multiplicity and F-signature of positive characteristic local rings. © 2013 Springer Science+Business Media New York. All rights reserved.
This book is the first of a two-volume basic introduction to enumerative combinatorics at a level suitable for graduate students and research mathematicians. It concentrates on the theory and application of generating functions, a fundamental tool in enumerative combinatorics. The book covers those parts of enumerative combinatorics of greatest applicability to other areas of mathematics. The four chapters are devoted to an introduction to enumeration (suitable for advanced undergraduates), sieve methods (including the Principle of Inclusion-Exclusion), partially ordered sets, and rational generating functions. There are a large number of exercises, almost all with solutions, which greatly augment the text and provide entry into many areas not covered directly. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.
The aim of this note is to determine the Hilbert-Kunz functions of rings defined by monomial ideals and of rings defined by
a single binomial equationX
a−Xb with gcd(X
a, Xb)=1.