scholarly journals On Buchsbaum rings obtained by gluing

1975 ◽  
Vol 83 ◽  
pp. 123-135 ◽  
Author(s):  
Shiro Goto
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Rees Algebra ◽  
Noetherian Local Ring ◽  
Parameter Ideal

Let A be a Noetherian local ring with maximal ideal m. In 1973 J. Barshay [1] showed that, if A is a Cohen-Macaulay ring, then so is the Rees algebra R(q) = ⊕n≧0qn for every parameter ideal q of A (cf. p. 93, Corollary). Recently the author and Y. Shimoda [5] have proved that the Rees algebra R(q) is a Cohen-Macaulay ring for every parameter ideal q of A if and only if(#) A is a Buchsbaum ring and for i ≠ 1, dim A.

scholarly journals On the Cohen-Macaulayfication of certain Buchsbaum rings

1980 ◽  
Vol 80 ◽  
pp. 107-116 ◽  
Author(s):  
Shiro Goto
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Local Ring ◽  
Parameter Ideal ◽  
The Difference ◽  
System Of Parameters

Let A be a Noetherian local ring of dimension d and with maximal ideal m. Then A is called Buchsbaum if every system of parameters is a weak sequence. This is equivalent to the condition that, for every parameter ideal q, the difference is an invariant I(A) of A not depending on the choice of q. (See Section 2 for the detail.) The concept of Buchsbaum rings was introduced by Stückrad and Vogel [8], and the theory of Buchsbaum singularities is now developing (cf. [6], [7], [9], [10], and [12]).

scholarly journals Reducibility of parameter ideals in low powers of the maximal ideal

2021 ◽  
pp. 181-193
Author(s):  
Katharine Shultis ◽  
Peder Thompson
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Upper Bounds ◽  
Gorenstein Rings ◽  
Content Type ◽  
Noetherian Local Ring ◽  
Parameter Ideal ◽  
Systems Of Parameters

A commutative noetherian local ring ( R , m ) (R,\mathfrak {m}) is Gorenstein if and only if every parameter ideal of R R is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer ℓ \ell (depending on R R ) such that R R is Gorenstein if and only if there exists an irreducible parameter ideal contained in m ℓ \mathfrak {m}^\ell . We give upper bounds for ℓ \ell that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Toward a theory of generalized Cohen-Macaulay modules

1986 ◽  
Vol 102 ◽  
pp. 1-49 ◽  
Author(s):  
Ngô Viêt Trung
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Finitely Generated ◽  
Noetherian Local Ring

Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = dim M≥1.

scholarly journals Weak Formal Schemes

1972 ◽  
Vol 45 ◽  
pp. 1-38 ◽  
Author(s):  
David Meredith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Local Ring ◽  
Formal Schemes

Throughout this paper, (R, m) denotes a (noetherian) local ring R with maximal ideal m.In [5], Monsky and Washnitzer define weakly complete R-algebras with respect to m. In brief, an R-algebra A† is weakly complete if

scholarly journals POWERS OF THE MAXIMAL IDEAL AND VANISHING OF (CO)HOMOLOGY

2020 ◽  
Vol 63 (1) ◽  
pp. 1-5
Author(s):  
OLGUR CELIKBAS ◽  
RYO TAKAHASHI
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Positive Power ◽  
Noetherian Local Ring ◽  
Torsion Condition

AbstractWe prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

Idealizations of maximal Buchsbaum modules over a Buchsbaum ring

1988 ◽  
Vol 104 (3) ◽  
pp. 451-478 ◽  
Author(s):  
Kikumichi Yamagishi
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring

Throughout this paper A denotes a Noetherian local ring with maximal ideal m and M denotes a finitely generated A-module. Moreover stands for the ith local cohomology functor with respect to m (cf. [10]). We refer to [15] for unexplained terminolog.

scholarly journals The structure of sally modules and Buchsbaumness of associated graded rings

2013 ◽  
Vol 212 ◽  
pp. 97-138 ◽  
Author(s):  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Associated Graded Rings

AbstractLet A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

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