New model structures and projective (injective) cotorsion pairs

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.

Remarks on Gorenstein Weak Injective and Weak Flat Modules

In this paper, we introduce Gorenstein weak injective and weak flat modules in terms of, respectively, weak injective and weak flat modules; the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules. In this new setting, we characterize rings over which all modules are Gorenstein weak injective. Moreover, we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module, and also the stability of Gorenstein weak injective modules.

A Note on DG-Gorenstein Injective Complexes

The notion of DG-Gorenstein injective complexes is studied in this article. It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with [Formula: see text] Gorenstein injective in R-Mod for each [Formula: see text] and any morphism [Formula: see text] is null homotopic whenever E is a DG-injective complex.

Acyclic Complexes and Gorenstein Rings

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

Recollements of stable categories of Gorenstein injective modules

Let [Formula: see text] and [Formula: see text] be rings, [Formula: see text] a [Formula: see text]-bimodule and [Formula: see text] be a triangular matrix ring. We first give an explicit description for Gorenstein injective [Formula: see text]-modules over the triangular matrix ring [Formula: see text], and then construct a (right) recollement of stable categories of Gorenstein injective modules.

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.