On Strongly Gorenstein FP-Injective Modules

This article continues to investigate a particular case of Gorenstein FP-injective modules, called strongly Gorenstein FP-injective modules. Some examples are given to show that strongly Gorenstein FP-injective modules lie strictly between FP-injective modules and Gorenstein FP-injective modules. Various results are developed, many extending known results in [1 Bennis , D. , Mahdou , N. ( 2007 ). Strongly Gorenstein projective, injective, and flat modules . J. Pure Appl. Algebra 210 : 437 – 445 .[Crossref], [Web of Science ®] , [Google Scholar]]. We also characterize FC rings in terms of strongly Gorenstein FP-injective, projective, and flat modules.     

Gorenstein AC-projective dimension of unbounded complexes

Recently, the notion of Gorenstein AC-projective (resp., Gorenstein AC-injective) modules was introduced in [3 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring. http://arxiv.org/abs/1405.5768. [Google Scholar]] by which the so-called “Gorenstein AC-homological algebra” was established. Here, we define and study a notion of Gorenstein AC-projective dimension for complexes (not necessarily bounded) over associative rings, which is inspired by Veliche’s construction of defining Gorenstein projective dimension. In particular, we show that such a dimension can be closely related to the “proper” Gorenstein AC-projective resolutions of complexes induced by a complete and hereditary cotorsion pair in the category of complexes of modules. This enables us to interpret this dimension of a complex in terms of vanishing of the derived functor RHom_R(?,?). As applications, some characterizations of the Gorenstein AC-projective dimension of a module are also obtained.     

On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

Gorenstein FP-Injective Dimension for Complexes

This article is a continuous work of [17 Hu , J. , Zhang , D. ( 2013 ). Weak AB-context for FP-injective modules with respect to semidualizing modules . J. Algebra Appl. 12 ( 7 ): 1350039 .[Crossref], [Web of Science ®] , [Google Scholar]], where the coauthors introduced the notion of 𝒢-FP-injective R-modules. In this article, we define a notion of 𝒢-FP-injective dimension for complexes over left coherent rings. To investigate the relationships between 𝒢-FP-injective dimension and FP-injective dimension for complexes, the complete cohomology group bases on FP-injectives is given.     

A Class of Gorenstein Artin Algebras of Embedding Dimension Four

In this article, we study height four graded Gorenstein ideals I in k[x, y, z, w] such that I ₂ is of height one and generated by three quadrics. After a suitable linear change of variables, I ∩ k[x, y, z] is either Gorenstein or of type two. The former case was studied by Iarrobino and Srinivasan [8 Iarrobino , A. , Srinivasan , H. ( 2005 ). Artininan Gorenstein algebras of embedding dimension four: components of ? Gor (H) for H = (1, 4, 7,…, 1) . Journal of Pure and Applied Algebra 201 : 62 – 96 .[Crossref], [Web of Science ®] , [Google Scholar]] where they give the structure of the ideal and its resolution. We study the latter case and give the structure of these ideals and their minimal resolution. We also explicitly write the form of the generators of I and the maps in the free resolution of R/I.     

GRADE AND GORENSTEIN DIMENSION

Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext ^t _R (M,R) ≠ 0, see [11] Rees, D. 1957. The Grade of an Ideal or Module. Proc. Camb. Phil. Soc., 53: 28–42. [Crossref] , [Google Scholar]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext ^t _R (M, R) ≠ 0, see [3] Auslander, M. 1967. Anneaux De Gorenstein Et Torsion En Algebre Commutative Edited by: Mangeney, M., Peskine, C. and Szpiro, L. Paris: Ecole Normale Superieure de Jeunes Filles.  [Google Scholar]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results.     

Cone Length for DG Modules and Global Dimension of DG Algebras

As the definition of free class of differential modules over a commutative ring in [1 Avramov , L. L. , Buchweitz , R.-O. , Iyengar , S. ( 2007 ). Class and rank of differential modules . Invent. Math. 169 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., A ^op -modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and D ^c (A) are not bigger than the DG free class of a minimal semifree resolution X of the DG A ^e -module A.     

Auslander–Reiten Triangles in Homotopy Categories

Let A be an artin algebra. We show that the bounded homotopy category of finitely generated right A-modules has Auslander–Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [14 Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar]]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules, admits Auslander–Reiten triangles, which improve a main result in [12 Nan, G. (2012). Auslander-Reiten triangles on Gorenstein derived categories. Comm. Algebra 40:3912–3919.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].