Journal of the
Korean Mathematical Society JKMS

Let $\mathcal{V}$, $\mathcal{W}$, $\mathcal{Y}$, $\mathcal{X}$ be four classes of left $R$-modules. The notion of $(\mathcal{V, W, Y, X})$-Gorenstein $R$-complexes is introduced, and it is shown that under certain mild technical assumptions on $\mathcal{V}$, $\mathcal{W}$, $\mathcal{Y}$, $\mathcal{X}$, an $R$-complex ${M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein if and only if the module in each degree of ${M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein and the total Hom complexs Hom$_R({Y},{M})$, Hom$_R({M},{X})$ are exact for any ${Y}\in\widetilde{\mathcal{Y}}$ and any ${X}\in\widetilde{\mathcal{X}}$. Many known results are recovered, and some new cases are also naturally generated.

Keywords: Gorenstein module, Gorenstein complex

MSC numbers: Primary 18G25, 18G35

Supported by: This work was financially supported by the National Natural Science Foundation of China (Grant nos. 11861055, 12061061 and 12361008) and Funds for Innovative Fundamental Research Group Project of Gansu Province (Grant no. 23JRRA684).