Strongly VW-Gorenstein N-complexes

Let V,W be two classes of R-modules. The notion of strongly VW-Gorenstein N-complexes is introduced, and under certain mild hypotheses on V and W, it is shown that an N-complex X is strongly VW-Gorenstein if and only if each term of X is a VW-Gorenstein module and N-complexes HomR(V,X) and HomR(X,W) are N-exact for any V and W. Furthermore, under the same conditions on V and W, it is proved that an N-exact N-complex X is VW-Gorenstein if and only if Zₙᵗ(X) is a VW-Gorenstein module for each n and each t=1,2,…,N-1. Consequently, we show that an N-complex X is strongly Gorenstein projective (resp., injective) if and only if X is N-exact and Zᵗₙ(X) is a Gorenstein projective (resp., injective) module for each n and t=1,2,…,N-1.