Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | Torm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This re...

A ring is called n-perfect (n ≥ 0), if every flat module has projective dimension less or equal than n. In this paper, we show that the n-perfectness relate, via homological approach, some homological dimension of rings. We study n-perfectness in some known ring constructions. Finally, several examples of n-perfect rings satisfying special conditions are given.

We study Gorenstein dimension and grade of a module M over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dimM ≤ G-dimgrM and an equality gradeM = grade grM , whe...

We define homological dimensions for S-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real K-theory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 2 or 3, and the global dimen...

A Gorenstein sequence H is a sequence of nonnegative integers H = (1, h1, . . . , hj = 1) symmetric about j/2 that occurs as the Hilbert function in degrees less or equal j of a standard graded Artinian Gorenstein algebra A = R/I , where R is a polynomial ring in r variables and I is a graded ideal. The scheme PGor(H) parametrizes all such Gorenstein algebra quotients of R having Hilbert functi...

Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | TorRm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This r...

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke’s theorem for infinite dimensional Hopf algebras. The generalization of Maschke’s theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

Let $(R,fm,k)$ be a local Gorenstein ring of dimension $n$. Let $H_{I,J}^i(R)$ be the  local cohomology with respect to a pair of ideals $I,J$ and $c$ be the $inf{i|H_{I,J}^i(R)neq0}$. A pair of ideals $I, J$ is called cohomologically complete intersection if $H_{I,J}^i(R)=0$ for all $ineq c$. It is shown that, when $H_{I,J}^i(R)=0$ for all $ineq c$, (i) a minimal injective resolution of $H_{I,...

In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize algebras finite dimension. As application, introduce almost n-precluster tilting establish a correspondence between modules n-minimal Auslander-Gorenstein algebras. Moreover, give description of Gorenstein projective over in terms corresponding modules.