𝐴-transforms and Hilbert functions in local lattices

Abstract

Introduction. In [1], Dilworth introduced the notion of a principal element of a multiplicative lattice and used it to define a class of lattices in which the Noether decomposition theorems hold.He called these Noether lattices and showed, among other things, that Noether lattices satisfy the Krull Intersection Theorem and the Krull Principal Ideal Theorem.Noether lattices in general, however, are lacking in many important ring-like properties.For example, in a local Noether lattice of altitude d, there need not exist d principal elements with join an element which is primary for the maximal element, so that systems of parameters may be much larger than one would hope.In this paper it is shown that if F is the maximal element of a local Noether lattice, then for large « the number of principal elements required to generate Fn is given by a polynomial B*(P, x) (Corollary 3.3).It seems reasonable then to consider the degree of this polynomial as a second measure of the number of elements in a system of parameters, or possibly as an alternative definition of the dimension of the lattice.Lemma 3.6 and Theorem 3.9 establish that the degree of B*(P, x) is d-1 where d is the height of F. §2 is concerned with the introduction of a graded multiplicative lattice, denoted 3&(IF, A).This lattice is analogous to the ^-transform of a Noetherian ring introduced by D. Rees and serves as the vehicle in §3 for obtaining the polynomial B*(P, x).The main result of this section is that, if "Sf is a Noether lattice, then âiï(FF, A) is a Noether lattice (Theorem 2.11).A determination is also made of the heights of the maximal primes of 3#(¿F, A) in terms of A and ¿F (Lemma 2.16).In §4 regular local lattices are considered.It is shown that the 0-element of a regular local lattice is prime (Theorem 4.1) and the results of §2 are applied to show that the degree function is a valuation (Corollary 4.4).The author expresses his gratitude to Professor Louis J. Ratliff for many helpful suggestions.Throughout the remainder of this paper, the symbol £F will denote a Noether lattice.1.The following lemma is a useful generalization of Lemma (6.3) in [1].Lemma 1.1.Let A, B and C be elements of IF with A principal, BS C and (0 : A) ACSB.Then the correspondence D-^AD of the quotient C\B into the quotient CA/BA is a lattice isomorphism.