Homological Dimensions Relative to Special Subcategories

Let A be an abelian category,(L) an additive,full and self-orthogonal subcategory of A closed under direct summands,rG((L)) the right Gorenstein subcategory of A relative to (L),and ⊥(L) the left orthogonal class of (L).For an object A in A,we prove that if A is in the right 1-orthogonal class of rG((L)),then the (L)-projective and rG((L))-projective dimensions of A are identical;if the rG((L))-projective dimension of A is finite,then the rG((L))-projective and ⊥(L)-projective dimensions of A are identical.We also prove that the supremum of the (L)-projective dimensions of objects with finite (L)-projective dimension and that of the rG((L))-projective dimensions of objects with finite rG((L))-projective dimension coincide.Then we apply these results to the category of modules.