We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory ω of an abelian category. We introduce the Frobenius category of ω-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of ω-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable cat...

Let [Formula: see text] be an abelian category. In this paper, we investigate the global text]-Gorenstein projective dimension text], associated to a GP-admissible pair text]. We give homological conditions over that characterize it. Moreover, given GI-admissible study under which and are same.

Let R be a commutative noetherian ring and Γ a finite group. In this paper,we study Gorenstein homological dimensions of modules with respect to a semi-dualizing module over the group ring  . It is shown that Gorenstein homological dimensions  of an  -RΓ module M with respect to a semi-dualizing module,  are equal over R and RΓ  .

Tate local cohomology and Gorenstein theory, which are important generalizations of the classical cohomology, has been investigated. It found that they have such vanishing properties long exact sequences. However, for homology, what about duality? In this paper we concerned with homology homology. first part generalize as study properties, artinianness some sequence modules. We find an artiania...

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...

For projective varieties with a certain class of ‘mild’ isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property (i.e. h st ≤ h p+1,q+1 st for p+ q ≤ d− 2, where d is the dimension of the variety). This result fits nicely with a 6-dimensional counterexample of Mustaţă and Pa...

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 ...

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

Let R be a right GF -closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorenstein ring R, the G...