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].

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 Local rings with quasi-decomposable maximal ideal

2018 ◽  
Vol 168 (2) ◽  
pp. 305-322 ◽  
Author(s):  
SAEED NASSEH ◽  
RYO TAKAHASHI
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Maximal Ideal ◽  
Projective Dimension ◽  
Complete Classification ◽  
Local Rings ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Direct Sum

AbstractLet (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 is a direct summand of (a direct sum of) syzygies of M. Applying this result to the case where 𝔪 is quasi-decomposable, we obtain several classifications of subcategories, including a complete classification of the thick subcategories of the singularity category of R.

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 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.

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.

Noetherian Dimension and Co-localization of Artinian Modules over Local Rings

2014 ◽  
Vol 21 (04) ◽  
pp. 663-670 ◽  
Author(s):  
Le Thanh Nhan ◽  
Tran Do Minh Chau
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Noetherian Dimension ◽  
Base Ring ◽  
Finitely Generated Modules

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dim RA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dim RA = dim (R/ Ann RA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.

On Special Fiber Rings of Modules

2019 ◽  
Vol 72 (1) ◽  
pp. 225-242
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Local Ring ◽  
Arbitrary Dimension ◽  
Residue Field ◽  
Independent Interest ◽  
General Situation ◽  
Finitely Generated ◽  
Fiber Ring ◽  
Noetherian Local Ring ◽  
Reduction Number

AbstractWe prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

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.

Certain countably generated big Cohen-Macaulay modules are balanced

1989 ◽  
Vol 105 (1) ◽  
pp. 73-77 ◽  
Author(s):  
R. Y. Sharp
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Commutative Algebra ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
System Of Parameters ◽  
Substantial Interest ◽  
Homological Conjectures

Throughout this note, A will denote a (commutative, Noetherian) local ring (with identity) having maximal ideal m and dimension d. Let x1, …, xd be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay A-module with respect to x1, …, xd, if x1, …, xd is an M-sequence. In the last ten or fifteen years there has been substantial interest in such modules, initially stemming from M. Hochster's discoveries that, if A contains a field as a subring, and x1, …,xd is any s.o.p. for A, then there exists a big Cohen-Macaulay A-module with respect to x1, …,xd, and that the existence of such modules has important consequences for the local homological conjectures in commutative algebra: see [6].

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.

Sign in / Sign up

Export Citation Format

Share Document