(𝒲, 𝒴, 𝒳)-Gorenstein complexes. Issue 7 (3rd July 2017) - Zhao, Renyu; Ding, Nanqing
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.