scholarly journals Cohen-Macaulay homological dimensions

2020 ◽  
Vol 126 (2) ◽  
pp. 189-208
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif ◽  
Siamak Yassemi

We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.

2017 ◽  
Vol 10 (03) ◽  
pp. 1750048
Author(s):  
Fatemeh Mohammadi Aghjeh Mashhad

Let [Formula: see text] be a local ring and [Formula: see text] denote the Matlis duality functor. Assume that [Formula: see text] possesses a normalized dualizing complex [Formula: see text] and [Formula: see text] and [Formula: see text] are two homologically bounded complexes of [Formula: see text]-modules with finitely generated homology modules. We will show that if G-dimension of [Formula: see text] and injective dimension of [Formula: see text] are finite, then [Formula: see text] Also, we prove that if Gorenstein injective dimension of [Formula: see text] and projective dimension of [Formula: see text] are finite, then [Formula: see text] These results provide some generalizations of Suzuki’s Duality Theorem and the Herzog–Zamani Duality Theorem.


2007 ◽  
Vol 35 (6) ◽  
pp. 1882-1889 ◽  
Author(s):  
Leila Khatami ◽  
Siamak Yassemi

2009 ◽  
Vol 137 (07) ◽  
pp. 2201-2207 ◽  
Author(s):  
Leila Khatami ◽  
Massoud Tousi ◽  
Siamak Yassemi

2019 ◽  
Vol 19 (09) ◽  
pp. 2050174
Author(s):  
Bo Lu ◽  
Zhenxing Di

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-complexes with [Formula: see text] an integer such that [Formula: see text] has finite Gorenstein projective dimension and [Formula: see text] has finite Gorenstein injective dimension. We define the [Formula: see text]th Gorenstein cohomology groups [Formula: see text] [Formula: see text] via a strict Gorenstein precover [Formula: see text] of [Formula: see text] and a strict Gorenstein preenvelope [Formula: see text] of [Formula: see text]. Using Gaussian binomial coefficients we show that there exists an isomorphism [Formula: see text] which extends the balance result of Liu [Relative cohomology of complexes. J. Algebra 502 (2018) 79–97] to the [Formula: see text]-complex case.


2007 ◽  
Vol 101 (1) ◽  
pp. 5 ◽  
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif

Ext-finite modules were introduced and studied by Enochs and Jenda. We prove under some conditions that the depth of a local ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of an Ext-finite module of finite Gorenstein injective dimension. Let $(R,\mathfrak m)$ be a local ring. We say that an $R$-module $M$ with $\dim_R M=n$ is a Grothendieck module if the $n$-th local cohomology module of $M$ with respect to $\mathfrak m$, $\mathrm{H}_{\mathfrak m}^n (M)$, is non-zero. We prove the Bass formula for this kind of modules of finite Gorenstein injective dimension and of maximal Krull dimension. These results are dual versions of the Auslander-Bridger formula for the Gorenstein dimension. We also introduce GF-perfect modules as an extension of quasi-perfect modules introduced by Foxby.


1999 ◽  
Vol 72 (2) ◽  
pp. 107-117 ◽  
Author(s):  
Edgar E. Enochs ◽  
Overtoun M.G. Jenda

2018 ◽  
Vol 61 (4) ◽  
pp. 865-877 ◽  
Author(s):  
Liran Shaul

AbstractLet A be a commutative noetherian ring, let a ⊆ A be an ideal, and let I be an injective A-module. A basic result in the structure theory of injective modules states that the A-module Γa(I) consisting of ɑ-torsion elements is also an injective A-module. Recently, de Jong proved a dual result: If F is a flat A-module, then the ɑ-adic completion of F is also a flat A-module. In this paper we generalize these facts to commutative noetherian DG-rings: let A be a commutative non-positive DG-ring such that H0(A) is a noetherian ring and for each i < 0, the H0(A)-module Hi(A) is finitely generated. Given an ideal ⊆ H0(A), we show that the local cohomology functor R associated with does not increase injective dimension. Dually, the derived -adic completion functor LΛ does not increase flat dimension.


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