Cohen-Macaulay homological dimensions

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.

Gorenstein cohomology of N-complexes

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.

Gorenstein homological dimensions and some duality results

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.

GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES

Given a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

Gorenstein Injective and Injective Complete Cohomological Dimensions of Groups

Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group Γ, called the injective complete cohomological dimension of Γ, denoted by iccd Γ. We study this dimension and investigate its properties. Also, we define the Gorenstein injective dimension of the group Γ, which is denoted by Gid Γ. We show that Gid Γ is related to iccd Γ, as well as to spli and silp invariants of Gedrich and Gruenberg. In particular, it is shown that iccd Γ is a refinement of Gid Γ. In addition, we show that silp Γ = spli Γ < ∞ if and only if the Shapiro lemma holds for injective complete cohomology.

GORENSTEIN INJECTIVE MODULES AND A GENERALIZATION OF ISCHEBECK FORMULA

Let (R,[Formula: see text]) be a commutative Noetherian local ring and let M and N be nonzero finitely generated R-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper, we prove a generalization of Ischebeck formula, that is [Formula: see text].