Zhaoyong Huang

Self-orthogonal modules over coherent rings

Abstract Let R be a left coherent ring, S any ring and R ω S an ( R , S )-bimodule. Suppose ω S has an ultimately closed FP-injective resolution and R ω S satisfies the conditions: (1) ω S is finitely presented; (2) The natural map R → End( ω S ) is an isomorphism; (3) Ext S i ( ω , ω )=0 for any i ≥1. Then a finitely presented left R -module A satisfying Ext R i ( A , ω )=0 for any i ≥1 implies that A is ω -reflexive. Let R be a left coherent ring, S a right coherent ring and R ω S a faithfully balanced self-orthogonal bimodule and n ≥0. Then the FP-injective dimension of R ω S is equal to or less than n as both left R -module and right S -module if and only if every finitely presented left R -module and every finitely presented right S -module have finite generalized Gorenstein dimension at most n .

Tilting modules of finite projective dimension and a generalization of ∗-modules ✩

Abstract It is well known that tilting modules of projective dimension ⩽ 1 coincide with ∗ -modules generating all injectives. This result is extended in this paper. Namely, we generalize ∗ -modules to so-called ∗ n -modules and show that tilting modules of projective dimension ⩽ n are ∗ n -modules which n-present all injectives.

Gorenstein Projective Dimension Relative to a Semidualizing Bimodule

Let S and R be rings and S C R a semidualizing bimodule. We investigate the relation between the G C -syzygy with the C-syzygy of a module as well as the relation between the G C -projective resolution and the projective resolution of a module. As a consequence, we get that if is an exact sequence of S-modules with all G i , G i G C -projective, such that Hom S (𝔾, T) is still exact for any module T which is isomorphic to a direct summand of direct sums of copies of S C, then Im(G 0 → G 0) is also G C -projective. We obtain a criterion for computing the G C -projective dimension of modules. When S C R is a faithfully semidualizing bimodule, we study the Foxby equivalence between the subclasses of the Auslander class and that of the Bass class with respect to C.

Selforthogonal modules with finite injective dimension II

Let Λ be a left and right Artin ring and ΛωΛ a faithfully balanced selforthogonal bimodule. We give a sufficient condition that the injective dimension of ωΛ is finite implies that of Λω is also finite.

The existence of maximal n-orthogonal subcategories

Abstract For an ( n − 1 ) -Auslander algebra Λ with global dimension n, we give some necessary conditions for Λ admitting a maximal ( n − 1 ) -orthogonal subcategory in terms of the properties of simple Λ-modules with projective dimension n − 1 or n. For an almost hereditary algebra Λ with global dimension 2, we prove that Λ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable Λ-module M, M is injective is equivalent to that the reduced grade of M is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence of maximal n-orthogonal subcategories of T ⊥ for a cotilting module T. For a Gorenstein algebra, we prove that all non-projective direct summands of a maximal n-orthogonal module are Ω n τ -periodic. In addition, we study the relation between the complexity of modules and the existence of maximal n-orthogonal subcategories for the tensor product of two finite-dimensional algebras.

Wt-approximation representations over quasik-Gorenstein algebras

The notions of quasik-Gorenstein algebras and Wt-approximation representations are introduced. The existence and uniqueness (up to projective equivalences) of Wt-approximation representations over quasi k-Gorenstein algebras are established. Some applications of Wt-approximation representations to homologically finite subcategories are given.

Duality of Preenvelopes and Pure Injective Modules

AbstractLet R be an arbitrary ring and (−) + = Hom Z (−,Q/Z) where Zis the ring of integersand Q is the ring of rational numbers, and let C be a subcategory of left R-modulesand D a subcategory of right R-modules such that X + ∈ D for any X ∈ C and allmodules in C are pure injective. Then a homomorphism f : A → C of left R-moduleswith C ∈ C is a C-(pre)envelope of A provided f + : C + → A + is a D-(pre)cover of A + .Some applications of this result are given. 1. IntroductionThroughout this paper, all rings are associative with identity. For a ring R, we use ModR(resp. ModR op ) to denote the category of left (resp. right) R-modules.(Pre)envelopes and (pre)covers of modules were introduced by Enochs in [E], and are fun-damental and important in relative homological algebra. Following Auslander and Smalφ’sterminology in [AuS], for a finitely generated module over an artinian algebra, a (pre)envelopeand a (pre)cover are called a (minimal) left approximation and a (minimal) right approxima-tion, respectively. Notice that (pre)envelopes and (pre)covers of modules are dual notions,so the dual properties between them are natural research topics. It has been known thatmost of their properties are indeed dual ([AuS, E, EH, EJ2, GT] and references therein).We write (−)

Torsionfree dimension of modules and self-injective dimension of rings

Let R be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated R-modules. For any n 0, we prove that R is a Gorenstein ring with self-injective dimension at most n if and only if every finitely generated left R-module and every finitely generated right R-module have torsionfree dimension at most n, if and only if every finitely generated left (or right) R-module has Gorenstein dimension at most n. For any n 1, we study the properties of the finitely generated R-modules M with Ext i R (M, R) 0 for any 1 i n. Then we investigate the relation between these properties and the s elf-injective dimension of R.

Selforthogonal Modules with Finite Injective Dimension III

Let R be a left Noetherian ring, S a right Noetherian ring and RU a generalized tilting module with S = End(RU). We give some equivalent conditions that the injective dimension of US is finite implies that of RU is also finite. As an application, under the assumption that the injective dimensions of RU and US are finite, we construct a hereditary and complete cotorsion theory by some subcategories associated with RU.

TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS

Letbe an Auslanders 1-Gorenstein Artinian algebra with global dimension two. Ifadmits a trivial maximal 1-orthogonal subcategory of mod�, then for any indecom- posable module M ∈ mod�, we have that the projective dimension of M is equal to one if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to two. In this case, we further get thatis a tilted algebra.