Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(Prj C(R)), is always well generated and is compactly generated p...

We define the symmetric Auslander category A(R) to consist of complexes of projective modules whose leftand righttails are equal to the leftand right-tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is th...

We introduce and investigate the notion of GC -projective modules over (possibly non-noetherian) commutative rings, where C is a semidualizing module. This extends Holm and Jørgensen’s notion of C-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite GC-projectiv...

In [Hov02], the second author introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the st...

we investigate the relative cohomology and relative homology theories of $f$-gorenstein modules, consider the relations between classical and $f$-gorenstein (co)homology theories.

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander’s theorem on alg...

Motivated by some properties satisfied Gorenstein projective and injective modules over an Iwanaga-Gorenstein ring, we present the concept of left right n-cotorsion pairs in abelian category C. Two classes A B objects C form a pair (A,B) if orthogonality relation ExtCi(A,B)=0 is for indexes 1≤i≤n, every object has resolution whose syzygies have B-resolution dimension at most n−1. This its dual ...

The category gp ( Λ ) of Gorenstein-projective modules over tensor algebra = A ⊗ k B can be described as the monomorphism mon , . In particular, Λ-modules are monic. this paper, we find similar relation between semi-Gorenstein-projective and -modules, via monic modules, namely, ⊥ ∩ Using this, it is proved that if weakly Gorenstein, then Gorenstein only each monic; Q with a finite acyclic quive...