(𝒲, 𝒴, 𝒳)-Gorenstein complexes. Issue 7 (3rd July 2017)

• Record Type: Journal Article
• Authors: Zhao, Renyu; Ding, Nanqing
• Is Part Of: Communications in algebra. Volume 45:Issue 7(2017)
• Page Start: 3075
• Page End: 3090
• Publication Date: 2017-07-03
• Subjects: (𝒲, 𝒴, 𝒳)-Gorenstein complex -- (𝒲, 𝒴, 𝒳)-Gorenstein module -- stability; 18G25 -- 18G35 -- 16E05 -- 16E30; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1235173
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let 𝒲, 𝒴, 𝒳 be three classes of left R -modules. In this paper, we introduce and study ( 𝒲, 𝒴, 𝒳 )-Gorenstein complexes as a common generalization of completely 𝒲 -resolved complexes [26 ], Gorenstein projective (resp., injective) complexes [8 ], Ding projective (resp., injective) complexes [32 ] and Gorenstein AC-projective (resp., AC-injective) complexes [4 ]. It is shown that under certain hypotheses, a complex C is ( 𝒲, 𝒴, 𝒳 )-Gorenstein if and only if each C n is a ( 𝒲, 𝒴, 𝒳 )-Gorenstein module and ℋ o m R ( Y, C ), ℋ o m R ( C, X ) are exact for any Y ∈ 𝒴 ̃, X ∈ 𝒳 ̃ . This result unifies the corresponding results of the aforementioned complexes. As applications, the stability of ( 𝒲, 𝒴, 𝒳 )-Gorenstein complexes and modules are explored.