Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras

Let A and B be Artin algebras and let M be an (A, B)-bimodule with M-A and M-B finitely generated. In this paper, we construct tilting pairs of subcategories and Wakamatsu tilting subcategories over an upper triangular matrix Artin algebra Lambda = ((A)(M)(0)(B)) using tilting pairs andWakamatsu tiling subcategories over A and B. Let C be a subcategory of A-mod and let D be a subcategory of B-mod. Consider the subcategory of left Lambda-modules B-D(C) = {((X)(Y))(f) : f is a monomorphism, Y is an element of D and Coker f is an element of C}. We prove the following results: (1) Assume that M circle times(B) T ' subset of T, M circle times(B) C ' subset of C and Tor(i)(B) (M, T ') = 0 for all i >= 1. Then (C, T) and (C ', T ') are n-tilting pairs respectively in A-mod and B-mod if and only if (B-C '(C), B-T '(T)) is an n-tilting pair in Lambda-mod. (2) Assume that M circle times(B) V subset of W and Tor(i)(B) (M, V-perpendicular to) = 0 for all i >= 1. If W and V are Wakamatsu tilting subcategories respectively in A-mod and B-mod, then B-V(W) is aWakamatsu tilting subcategory in Lambda-mod.