(𝒲,𝒴,𝒳)-Gorenstein complexes

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 Cⁿ is a (𝒲,𝒴,𝒳)-Gorenstein module and $\mathcal{\mathscr{H}}{om}_R(Y,C),\mathcal{\mathscr{H}}{om}_R(C,X)$ are exact for any $Y\in \widetilde{𝒴},X\in \widetilde{𝒳}$. This result unifies the corresponding results of the aforementioned complexes. As applications, the stability of (𝒲,𝒴,𝒳)-Gorenstein complexes and modules are explored.

KEYWORDS:

• (𝒲,𝒴,𝒳)-Gorenstein complex
• (𝒲,𝒴,𝒳)-Gorenstein module
• stability

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 18G25
• 18G35
• 16E05
• 16E30

Acknowledgments

The authors are grateful to the anonymous referee for the careful reading of this article and many useful suggestions. This research was partially supported by the National Natural Science Foundation of China (11371187, 11361051, 11361052).