Proper resolutions and Gorensteinness in extriangulated categories

摘要

Let (ℓ, E, s) be an extriangulated category with a proper class ξ of E-triangles, and W an additive full subcategory of (ℓ, E,s). We provide a method for constructing a proper W (ξ)-resolution (resp., coproper W(ξ)-coresolution) of one term in an E-triangle in ξ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category g 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.