scholarly journals When is $R \ltimes I$ an almost Gorenstein local ring?

2017 ◽  
Vol 146 (4) ◽  
pp. 1431-1437 ◽  
Author(s):  
Shiro Goto ◽  
Shinya Kumashiro
Keyword(s):  
Local Ring ◽  
Gorenstein Local Ring

A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory

2017 ◽  
Vol 120 (2) ◽  
pp. 161 ◽  
Author(s):  
Tony J. Puthenpurakal
Keyword(s):  
Local Ring ◽  
Representation Type ◽  
Gorenstein Ring ◽  
Stable Category ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Isomorphism Classes ◽  
Liaison Theory ◽  
Gorenstein Local Ring ◽  
Infinite Set

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\operatorname{\underline{CM}}(A)$ be the stable category of maximal Cohen-Macauley $A$-modules and let $\operatorname{\underline{ICM}}(A)$ denote the set of isomorphism classes in $\operatorname{\underline{CM}}(A)$. We define a function $\xi \colon \operatorname{\underline{ICM}}(A) \to \mathbb{Z}$ which behaves well with respect to exact triangles in $\operatorname{\underline{CM}}(A)$. We then apply this to (Gorenstein) liaison theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liaison classes of $\{\,\mathfrak{m}^n \mid n\geq 1 \,\}$ is an infinite set. We also prove that if $A$ is Henselian with finite representation type with $A/\mathfrak{m}$ uncountable then for each $m \geq 1$ the set $\mathcal {C}_m = \{\, I \mid I \text { is a codim $2$ CM-ideal with } e_0(A/I) \leq m \,\}$ is contained in finitely many even liaison classes $L_1,\dots ,L_r$ (here $r$ may depend on $m$).

On the cofiniteness of Artinian local cohomology modules

2016 ◽  
Vol 15 (04) ◽  
pp. 1650070 ◽  
Author(s):  
Ghader Ghasemi ◽  
Kamal Bahmanpour ◽  
Jafar A’zami
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and [Formula: see text] an ideal of [Formula: see text]. Let [Formula: see text] be a nonzero finitely generated [Formula: see text]-module and [Formula: see text] be an integer. In this paper we show that, the [Formula: see text]-module [Formula: see text] is nonzero and [Formula: see text]-cofinite if and only if [Formula: see text]. Also, several applications of this result will be included.

scholarly journals Quasi-socle ideals in a Gorenstein local ring

2008 ◽  
Vol 212 (5) ◽  
pp. 969-980 ◽  
Author(s):  
Shiro Goto ◽  
Ryo Takahashi ◽  
Naoyuki Matsuoka
Keyword(s):  
Local Ring ◽  
Gorenstein Local Ring

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 95-101
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Injective Resolution ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Gorenstein Injective ◽  
Cousin Complexes ◽  
Cohomology Modules ◽  
Cousin Complex

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

On Weakly Laskerian and Weakly Cofinite Modules

2012 ◽  
Vol 19 (04) ◽  
pp. 693-698
Author(s):  
Kazem Khashyarmanesh ◽  
M. Tamer Koşan ◽  
Serap Şahinkaya
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cofinite Modules ◽  
Gorenstein Local Ring

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

Complete Cohomologies and Some Homological Invariants

2007 ◽  
Vol 14 (01) ◽  
pp. 155-166
Author(s):  
Javad Asadollahi ◽  
Shokrollah Salarian
Keyword(s):  
Local Ring ◽  
Sufficient Conditions ◽  
Cohomology Theory ◽  
Local Rings ◽  
Sufficient Condition ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Loewy Length ◽  
Gorenstein Local Ring

There is a complete cohomology theory developed over a commutative noetherian ring in which injectives take the role of projectives in Vogel's construction of complete cohomology theory. We study the interaction between this complete cohomology, that is referred to as [Formula: see text]-complete cohomology, and Vogel's one and give some sufficient conditions for their equivalence. Using [Formula: see text]-complete functors, we assign a new homological invariant to any finitely generated module over an arbitrary commutative noetherian local ring, that would generalize Auslander's delta invariant. We generalize the results about the δ-invariant to arbitrary rings and give a sufficient condition for the vanishing of this new invariant. We also introduce an analogue of the notion of the index of a Gorenstein local ring, introduced by Auslander, for arbitrary local rings and study its behavior under flat extensions of local rings. Finally, we study the connection between the index and Loewy length of a local ring and generalize the main result of [11] to arbitrary rings.

scholarly journals On Vanishing of Generalized Local Cohomology Modules

2005 ◽  
Vol 12 (02) ◽  
pp. 213-218 ◽  
Author(s):  
K. Divaani-Aazar ◽  
R. Sazeedeh ◽  
M. Tousi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

Let [Formula: see text] denote an ideal of a d-dimensional Gorenstein local ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text].

Some loci related to Cohen–Macaulayness

2014 ◽  
Vol 13 (06) ◽  
pp. 1450021
Author(s):  
Nguyen Thi Kieu Nga
Keyword(s):  
Local Ring ◽  
Finitely Generated ◽  
Canonical Module ◽  
Noetherian Local Ring ◽  
Gorenstein Local Ring

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. The pseudo Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness) was introduced by Cuong–Nhan [Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules, J. Algebra267 (2003) 156–177] as an extension of the Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness). In this paper, we describe the pseudo Cohen–Macaulay (pseudo CM) locus and pseudo generalized Cohen–Macaulay (pseudo generalized CM) locus of M. We also study the non-Cohen–Macaulay locus and the non-generalized Cohen–Macaulay locus of the canonical module K(M) of M in case where R is a quotient of a Gorenstein local ring.

Sign in / Sign up

Export Citation Format

Share Document