Gorenstein conditions over triangular matrix rings

Gorenstein homological dimensions of modules over triangular matrix rings

“…(1) Suppose that M 1 is strongly Gorenstein projective and let P : · · · → P f → P f → P → · · · be an exact sequence consisting of projective left A-modules, which is Hom B (−, C)-exact for each projective left A-module C and such that Ker∂ 0 ∼ = M 1 . By [5,Lemma 2.3], we get that the complex U ⊗ A P is exact in A-Mod, which implies that the complex p(P) is exact in T -Mod. And it clearly verifies that Ker∂ 0 p(P) = p(M 1 , 0).…”

“…Then p(P) is an exact sequence of left T -modules such that Ker∂ 0 ∼ = p(0, M 2 ). It remains to see that it is Hom T ( , C)-exact for each projective left T -module C. Let C be a projective left T -module, and note that, as a consequence of [5,Corollary 2.3], there exists a projective object (C 1 , C 2 ) in A-Mod×B-Mod such that p(C 1 , C 2 ) = C. Then, C = C1 (U⊗C1)⊕C2 . Now, using adjointness, we get that the complex Hom T (p(P), C) is isomorphic to the complex Hom B (P, U ⊗ C 1 ) ⊕ Hom B (P, C 2 ).…”

Recollements associated to cotorsion pairs over upper triangular matrix rings

“…Denote by ψ the corresponding homomorphism from Y to Hom A (M, X) by adjoint isomorphism. Some important classes of modules over upper triangular matrix rings have been studied by many authors (see, e.g., [21]- [22], [38]- [39] and [10], and their references). Now we recall the characterizations of the following classes of left T -modules.…”

“…(4) ([10, Theorem 3.5]) Suppose that A M has finite projective dimension, M B has finite flat dimension and A is left Gorenstein regular (see [10,Definition 2.1] or Section 4 below). Then X is Gorenstein projective if and only if X 2 and coker φ are Gorenstein projective and the homomorphism φ is monomorphic.…”

Homological dimension formulas for trivial extension algebras

“…However, the methods "multiple Tor" for the Palmer-Roos-Löfwall formula was so complicated that the formula has never got attention which it ought to deserve. For instance, in the studies of homological dimensions of upper triangular algebras (e.g., [2,9,11,34]) there have been no attempt to generalize the formula in such a way as to be applicable for each problem.…”