Complete Cohomology for Extriangulated Categories

Let [Formula: see text] be an extriangulated category with a proper class [Formula: see text] of [Formula: see text]-triangles. We study complete cohomology of objects in [Formula: see text] by applying [Formula: see text]-projective resolutions and [Formula: see text]-injective coresolutions constructed in [Formula: see text]. Vanishing of complete cohomology detects objects with finite [Formula: see text]-projective dimension and finite [Formula: see text]-injective dimension. As a consequence, we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein. Moreover, we give a general technique for computing complete cohomology of objects with finite [Formula: see text]-[Formula: see text]projective dimension. As an application, the relations between [Formula: see text]-projective dimension and [Formula: see text]-[Formula: see text]projective dimension for objects in [Formula: see text] are given.

The Category of Modules on an n-Trivial Extension: the Basic Properties

In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules. Namely, we introduce the right (resp., left) n-trivial extension of a category by a family of endofunctors. Among other results, projective, injective and flat objects of this category are characterized, and two applications are presented at the end of this paper. We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring.

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.

Behavior of the Auslander condition with respect to regradings

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen–Macaulay property from the same perspective as that for the Auslander condition. A key step of our approach is to establish homological relations between a graded ring [Formula: see text], its quotient ring modulo the ideal [Formula: see text] and its localization ring with respect to the Ore set [Formula: see text], where [Formula: see text] is a homogeneous regular normal non-invertible element of [Formula: see text].