Proper resolutions and Gorensteinness in extriangulated categories

Abstract

Let ($\mathscr{C},\mathbb{E},\frak{s}$) be an extriangulated category with a proper class ξ of $\mathbb{E}$-triangles, and $\mathscr{W}$ an additive full subcategory of ($\mathscr{C},\mathbb{E},\frak{s}$). We provide a method for constructing a proper $\mathscr{W}$(ξ)-resolution (resp., coproper $\mathscr{W}$(ξ)-coresolution) of one term in an $\mathbb{E}$-triangle in ξ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category $\mathscr{G}\mathscr{W}$(ξ) in extriangulated categories. These results generalize the work of Z. Y. Huang [J. Algebra, 2013, 393: 142–169] and X. Y. Yang and Z. C. Wang [Rocky Mountain J. Math., 2017, 47: 1013–1053], but the proof is not too far from their case. Finally, we give some applications about our main results.