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...

Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems on existence and uniqueness of Picard-Vessiot rings, as well as on Galois groups corresponding to the Picard-Vessiot rings. As the present approach restrict...

In this paper we examine the role of four Hilbert functions in the determination of the defining relations of the Rees algebra of almost complete intersections of finite colength. Because three of the corresponding modules are Artinian, some of these relationships are very effective, opening up tracks to the determination of the equations and also to processes of going from homologically define...

Let R = ⊕ n≥0 Rn be a homogeneous Noetherian ring with local base ring (R0,m0) and let M be a finitely generated graded R-module. Let Hi R+ (M) be the i-th local cohomology module of M with respect to R+ := ⊕ n>0 Rn. If dimR0 ≤ 1, the R-modules Γm0R(H R+ (M)), (0 :Hi R+ (M) m0) and Hi R+ (M)/m0H i R+ (M) are Artinian for all i ∈ N0. As a consequence, much can be said on the asymptotic behaviour...

Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standardgraded ring, $M$ be a finitely generated graded $R$-module and $J$be a homogenous ideal of $R$. In this paper we study the gradedstructure of the $i$-th local cohomology module of $M$ defined by apair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. Moreprecisely, we discuss finiteness property and vanishing of thegraded components $H^...