記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：TAYLOR & FRANCIS INC  

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.

DOI： 10.1080/00927872.2013.774407

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：AMER MATHEMATICAL SOC  

Let R be a Noetherian N-r-graded ring generated ill degrees d(1), ... , d(r) which are linearly independent vectors over R, and let a be an ideal in R-0. In this paper, we investigate the asymptotic behavior of the grade of the ideal a on the homogeneous components M-n of a finitely generated Z(r)-graclecl R-module M and show that the periodicity occurs in a cone.

DOI： 10.1090/S0002-9939-2011-11370-4

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In this article, we prove that the Buchsbaum-Rim function l(A)(S(nu+1) (F)/N(nu+1)) of a parameter module N in F is bounded above by e(F/N)((nu+d+r-1)(d+r-1)) for every integer nu >= 0. Moreover, it turns out that the base ring A is Cohen-Macaulay once the equality holds for some integer nu. As a direct consequence, we observe that the first Buchsbaum-Rim coefficient e(1)(F/N) of a parameter module N is always non-positive. (C) 2010 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2010.09.035

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：AMER MATHEMATICAL SOC  

In this article we prove that the Buchsbaum-Rim multiplicity e(FIN) 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.

DOI： 10.1090/S0002-9939-09-10119-3

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：MATH SOC JAPAN  

Let R be a Noetherian local ring with the maximal ideal m and dim R = 1. In this paper, we shall prove that the module ExtR(R)(1)(R/Q,R) does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m(2) kills the 0th local cohomology module H-m(0)(R). The assertion is no longer true unless v(R) <= 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.

DOI： 10.2969/jmsj/06041045

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：TAYLOR & FRANCIS INC  

In this article we introduce a certain family of graded modules associated to a given module. These modules provide a natural extension of the notion of the associated graded ring of an ideal. We will investigate their properties. In particular, we will try to extend Ree' theorem on the associated graded ring of an ideal generated by a regular sequence to this context.

DOI： 10.1080/00927870802177291

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let A subset of B be a homogeneous extension of Noetherian standard N-r-graded rings with A(0) = B-0 = R. Let M be a finitely generated N-r-graded B-module and N subset of M a finitely generated graded A-submodule of M. In this paper, we investigate the asymptotic behavior of the set of primes associated to the module M-n/N-n and prove that for all sufficiently large n epsilon N-r, the set Ass(R)(M-n/N-n) is stable. We also give a certain inequality for the spread of standard multigraded rings, which is a natural generalization of Burch's inequality for the analytic spread of an ideal. (c) 2006 Published by Elsevier Inc.

DOI： 10.1016/j.jalgebra.2006.03.020

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記述言語：英語   掲載種別：研究論文（国際会議プロシーディングス）   出版者・発行元：AMER MATHEMATICAL SOC  

Let (A, m) be a regular local ring and let M subset of F = A(d) be the second syzygy module of the residue field A/m. In this paper, we introduce the concept of generalized Grassmann algebras and investigate the Rees algebra R(M) of M. We explore the properties of generalized Grassmann algebras and prove that the Rees algebra R(M) of the second syzygy module M is a Gorenstein factorial domain.

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記述言語：英語   掲載種別：研究論文（学術雑誌）  

Let A be a regular local ring and let ℱ = {Fn}n∈Zdbl
be a filtration of ideals in A such that ℛ(ℱ) = ⊕n≥0 Fn is a Noetherian ring with dim ℛ(ℱ) dim A + 1. Let script G sign (ℱ) = ⊕n≥0 Fn/Fn+1 and let a(script G sign(ℱ)) be the a-invariant of script G sign(ℱ). Then the theorem says that F1 is a principal ideal and Fn = F1n for all n ∈ Zdbl
if and only if script G sign (ℱ) is a Gorenstein ring and a script G sign(ℱ)) = -1. Hence a script G sign(ℱ)) ≤ -2, if script G sign (ℱ) is a Gorenstein ring, but the ideal F1 is not principal.

DOI： 10.1090/S0002-9939-02-06635-2

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記述言語：英語   掲載種別：研究論文（国際会議プロシーディングス）   出版者・発行元：SPRINGER  

Let A be a Noetherian local ring with the maximal ideal m and d = dim A. The set X-A of Gorenstein m-primary integrally closed ideals in A is explored in this paper. If k = A/m is algebraically closed and d greater than or equal to 2, then X-A is infinite. In contrast, for each field k which is not algebraically closed and for each integer d greater than or equal to 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) X-A = {m}, and (3) k = A/m. When d = 1, X-A is finite if and only if (A) over cap /p is not a DVR for any p is an element of Min (A) over cap, where (A) over cap denotes the m-adic completion. The list of elements in X-A is given, when A is a one-dimensional Noetherian complete local integral domain.

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：AMER MATHEMATICAL SOC  

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

DOI： 10.1090/S0002-9939-02-06436-5

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記述言語：英語   会議種別：口頭発表（一般）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：日本語  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（一般）  

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記述言語：英語   会議種別：口頭発表（一般）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：日本語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：日本語  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語  

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記述言語：英語  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（一般）  

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