Let \(({\mathcal {C}},{\mathbb {E}},{\mathfrak {s}})\) be an extriangulated category with a proper class \(\xi \) of \({\mathbb {E}}\)-triangles. In a previous work, we introduced and studied the \(\xi \)-\({\mathcal {G}}\)projective and the \(\xi \)-\({\mathcal {G}}\)injective dimension for any object in \({\mathcal {C}}\). In this paper, we first give some characterizations of \(\xi \)-\({\mathcal {G}}\)projective dimension by using derived functors on \({\mathcal {C}}\). Consequently, we show that the following equality holds under some assumptions:

where \(\xi \text {-}{\mathcal {G}}\mathrm{pd}M\) (resp., \(\xi \text {-}{\mathcal {G}}\mathrm{id}M\)) denotes the \(\xi \)-\({\mathcal {G}}\)projective dimension (resp., \(\xi \)-\({\mathcal {G}}\)injective dimension) of M. As an application, our main results generalize the work by Bennis–Mahdou and Ren–Liu, which are new for an exact category case. Moreover, our proof is far from the usual module or triangulated case.

让\（（{\ mathcal {C}}，{\ mathbb {E}}，{\ mathfrak {S}}）\）是extriangulated类别以适当的类\（\ XI \）的\（{\ mathbb {E}} \）-三角形。在以往的工作中，我们提出并研究了\（\ XI \） - \（{\ mathcal {G}} \）投影和\（\ XI \） - \（{\ mathcal {G}} \）内射\（{\ mathcal {C}} \）中任何对象的尺寸。在本文中，我们首先使用\（{\ mathcal {C}} \）上的派生函子来给出\（\ xi \） - \（{\ mathcal {G}} \）投影维的一些刻画。因此，我们表明在某些假设下，以下等式成立：

其中\（\ xi \ text {-} {\ mathcal {G}} \ mathrm {pd} M \）（分别是\（\ xi \ text {-} {\ mathcal {G}} \ mathrm {id} M \））表示\（\ xi \） - \（{\ mathcal {G}} \）射影维（resp。，\（\ xi \） - \（{\ mathcal {G}} \）内射维）的中号。作为应用，我们的主要结果概括了Bennis–Mahdou和Ren–Liu的工作，这对于确切的类别案例来说是新的。而且，我们的证明与通常的模块或三角案例相去甚远。