(𝒲,𝒴,𝒳)-Gorenstein complexes

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 Xin, D. W., Chen, J. L., Zhang, X. X. (2013). Completely 𝒲-resolved complexes. Commun. Algebra 41(3):1094–1106.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], Gorenstein projective (resp., injective) complexes [8 Enochs, E. E., García Rozas, J. R. (1998). Gorenstein injective and projective complexes. Commun. Algebra 26(5):1657–1674.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], Ding projective (resp., injective) complexes [32 Yang, G., Liu, Z. K., Liang, L. (2013). Model structures on categories of complexes over Ding-Chen rings. Commun. Algebra 41(1):50–69.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and Gorenstein AC-projective (resp., AC-injective) complexes [4 Bravo, D., Gillespie, J. (2016). Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44(5):2213–2233.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. 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 ℋomR(Y,C),ℋomR(C,X) are exact for any Y∈𝒴̃,X∈𝒳̃. This result unifies the corresponding results of the aforementioned complexes. As applications, the stability of (𝒲,𝒴,𝒳)-Gorenstein complexes and modules are explored.