Generalized fractions, Buchsbaum modules and generalized Cohen-Macaulay modules

1985 ◽  
Vol 98 (3) ◽  
pp. 429-436 ◽  
Author(s):  
R. Y. Sharp ◽  
H. Zakeri
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Positive Dimension ◽  
Noetherian Local Ring ◽  
Systems Of Parameters

Let A be a (commutative Noetherian) local ring (with identity) having maximal ideal m and positive dimension. This note is concerned with, among other things, a complex of A -modules which was studied in [8] and which involves modules of generalized fractions derived from A and subsets of systems of parameters for A; in ([8], 3·5), the complex was shown to have connections with local cohomology. The complex is described as follows.

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 Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

2010 ◽  
Vol 199 ◽  
pp. 95-105 ◽  
Author(s):  
Shiro Goto ◽  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Local Rings ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients

AbstractLet (A,m) be a Noetherian local ring withd= dimA≥ 2. Then, ifAis a Buchsbaum ring, the first Hilbert coefficientsofAfor parameter idealsQare constant and equal towherehi(A)denotes the length of theith local cohomology moduleofAwith respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves thatAis a Buchsbaum ring ifAis unmixed and the valuesare constant, which are independent of the choice of parameter idealsQinA. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

scholarly journals Krull dimension and generalized fractions

1985 ◽  
Vol 28 (3) ◽  
pp. 349-353 ◽  
Author(s):  
M. A. Hamieh ◽  
R. Y. Sharp
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Image Position ◽  
System Of Parameters

Let R be a (commutative Noetherian) local ring (with identity) having maximal ideal and dimension d≧l. It is shown in [5,3.6rsqb; that the local cohomology module may be described as a module of generalized fractions: if x1…,xd is a system of parameters for R, then , where U(x)d+1 is the triangular subset [4,2.1] of Rd+1 given by

scholarly journals Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

2010 ◽  
Vol 199 ◽  
pp. 95-105 ◽  
Author(s):  
Shiro Goto ◽  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Local Rings ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients

AbstractLet (A,m) be a Noetherian local ring with d = dim A ≥ 2. Then, if A is a Buchsbaum ring, the first Hilbert coefficients of A for parameter ideals Q are constant and equal to where hi(A) denotes the length of the ith local cohomology module of A with respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves that A is a Buchsbaum ring if A is unmixed and the values are constant, which are independent of the choice of parameter ideals Q in A. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

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.

Cofiniteness and vanishing of local cohomology modules

1991 ◽  
Vol 110 (3) ◽  
pp. 421-429 ◽  
Author(s):  
Craig Huneke ◽  
Jee Koh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Residue Field ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a noetherian local ring with maximal ideal m and residue field k. If M is a finitely generated R-module then the local cohomology modules are known to be Artinian. Grothendieck [3], exposé 13, 1·2 made the following conjecture:If I is an ideal of R and M is a finitely generated R-module, then HomR (R/I, ) is finitely generated.

scholarly journals FROBENIUS ACTIONS ON LOCAL COHOMOLOGY MODULES AND DEFORMATION

2017 ◽  
Vol 232 ◽  
pp. 55-75 ◽  
Author(s):  
LINQUAN MA ◽  
PHAM HUNG QUY
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Nonzero Divisor ◽  
Cohomology Modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzero divisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.

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

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