Futoshi Hayasaka

Finite homological dimension and primes associated to integrally closed ideals

Let I be an integrally closed ideal in a commutative Noetherian ring A. Then the local ring Ap is regular (resp. Gorenstein) for every p E Ass A A/I if the projective dimension of I is finite (resp. the Gorenstein dimension of I is finite and A satisfies Serres condition (S 1 )).

The a-invariant and gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let A be a regular local ring and let F = {F n } n ∈ z be a filtration of ideals in A such that R(F) = ○+ n>0 F n is a Noetherian ring with dimR(F) = dimA + 1. Let G(F) = ○+ n>0 F n /F n+1 and let a(G(F)) be the a-invariant of G(F). Then the theorem says that F 1 is a principal ideal and F n = F n 1 for all n ∈ Z if and only if G(F) is a Gorenstein ring and a(G(F)) = -1. Hence a(G(F)) ≤ -2, if G(F) is a Gorenstein ring, but the ideal F 1 is not principal.

A note on the Buchsbaum-Rim multiplicity of a parameter module

In this article we prove that the Buchsbaum-Rim multiplicity e(F/N) of a parameter module N in a free module F = A r is bounded above by the colength l A (F/N). Moreover, we prove that once the equality l A (F/N) = e(F/N) holds true for some parameter module N in F, then the base ring A is Cohen-Macaulay.

Asymptotic periodicity of grade associated to multigraded modules

Let R be a Noetherian Nr-graded ring generated in degrees d1, . . . ,dr which are linearly independent vectors over R, and let a be an ideal in R0. In this paper, we investigate the asymptotic behavior of the grade of the ideal a on the homogeneous components Mn of a finitely generated Zr-graded R-module M and show that the periodicity occurs in a cone.

Asymptotic Periodicity of Primes Associated to Multigraded Modules

In this paper, we investigate the asymptotic behavior of the set of primes associated to a graded ring extension of Noetherian multigraded rings and modules, and prove that the periodicity occurs in a cone. We also prove the same asymptotic behavior of the grade. The previous known results on this subject are recovered as a special case.

Towards a Theory of Gorenstein

Let A be a Noetherian local ring with the maximal ideal m and d = dim A. The set Χ A of Gorenstein m-primary integrally closed ideals in A is explored in this paper. If k = A/m is alge- braically closed and d≥2, then ΧA is infinite. In contrast, for each field k which is not algebraically closed and for each integer d ≥ 0, there exists a Noetherian complete equi-characteristic local integral domain A with dim A = d such that (1) the normalization of A is regular, (2) ΧA = {m}, and (3) k=A/m. When d = 1, ΧA is finite if and only if A/p is not a DVR for any p E Min A, where A denotes the m-adic completion. The list of elements in ΧA is given, when A is a one-dimensional Noetherian complete local integral domain.