Strongly $\mathcal{VW}$-Gorenstein $N$-complexes

Abstract

Let $\mathcal{V},\mathcal{W}$ be two classes of $R$-modules. The notion of strongly $\mathcal{VW}$-Gorenstein $N$-complexes is introduced, and under certain mild hypotheses on $\mathcal{V}$ and $\mathcal{W}$, it is shown that an $N$-complex $X$ is strongly $\mathcal{VW}$-Gorenstein if and only if each term of $X$ is a $\mathcal{VW}$-Gorenstein module and $N$-complexes ${\rm Hom}_{R}(V,X)$ and $ Hom_{R}(X,W)$ are $N$-exact for any $V\in\mathcal{V}$ and $W\in\mathcal{W}$. Furthermore, under the same conditions on $\mathcal{V}$ and $\mathcal{W}$, it is proved that an $N$-exact $N$-complex $X$ is $\mathcal{VW}$-Gorenstein if and only if $\mathrm{Z}_{n}^{t}(X)$ is a $\mathcal{VW}$-Gorenstein module for each $n\in\mathbb{Z}$ and each $t=1,2,\ldots,N-1$. Consequently, we show that an $N$-complex $X$ is strongly Gorenstein projective (resp., injective) if and only if $X$ is $N$-exact and ${\rm Z}^t_{n}(X)$ is a Gorenstein projective (resp., injective) module for each $n\in\mathbb{Z}$ and $t=1,2,\ldots,N-1$.