Let φ : (R, m)→ (S, n) be a local homomorphism of commutative noetherian local rings. Suppose that M is a finitely generated S-module. A generalization of Grothendieck’s non-vanishing theorem is proved for M (i.e. the Krull dimension of M over R is the greatest integer i for which the ith local cohomology module of M with respect to m, Hi m(M), is non-zero). It is also proved that the Gorenstei...

Let Λ be an Artinian algebra and F an additive subbifunctor of ExtΛ(−,−) having enough projectives and injectives. We prove that the dualizing subvarieties of mod Λ closed under F -extensions have F -almost split sequences. Let T be an F -cotilting module in mod Λ and S a cotilting module over Γ = End(T ). Then Hom(−, T ) induces a duality between F -almost split sequences in ⊥F T and almost sp...

We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it and reflects give conditions on when a stable objects, singularity defect categories, respectively. In appendix, we direct proof following known result: for an category with enough projectives injectives, its global coincides injective dimension.

In an unpublished work of Fasel–Rao–Swan the notion relative Witt group WE(R,I) is defined. this paper, we will give details construction. Then study injectivity Vaserstein symbol VR,I:Um3(R,I)/E3(R,I)→WE(R,I). We establish if R a non-singular affine algebra dimension 3 over perfect C1-field and I local complete intersection ideal R. It natural to ask whether injective for singular 3-dimensiona...

Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, sec...

Let R be a local ring of positive characteristic and X complex with nonzero finitely generated homology finite injective dimension. We prove that if the derived base change via Frobenius (or more generally, contracting) endomorphism has dimension then is Gorenstein. In particular, we give an affirmative answer to question by Falahola Marley [7, Question 3.9].

Let $H$ be a weak Hopf algebra that is finitely generated module over its affine center. We show has finite self-injective dimension and so the Brown--Goodearl Conjecture holds in this special setting.

Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen–Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, w...

In this paper we present some new properties of the metric dimension defined by Bouligand in 1928 and prove the following new projection theorem: Let dimb(A − A) denote the Bouligand dimension of the set A − A of differences between elements of A. Given any compact set A ⊆ R such that dimb(A−A) < m, then almost every orthogonal projection P of A of rank m is injective on A and P |A has Lipschit...

In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dime...