Relative homological dimensions in recollements of extriangulated categories

Let (A,B, C) be a recollement of extriangulated categories with a negative first extension and X-B be a generative subcategory of B. We prove that A and C have a generative subcategory, respectively. Using these subcategories, we obtain proper classes of E-triangles. Moreover, we compare A,B and C with respect to the xi -global dimensions of B under mild conditions. In particular, we show that the finiteness of the xi -global dimensions of extriangulated categories is invariant with respect to the recollements of extriangulated categories. As an application, these results generalize the work by Hu-Yao in triangulated categories.