Homological theory of recollements of abelian categories

The Extension Dimension of Abelian Categories

“…Let Λ, Λ ′ , Λ ′′ be artin algebras and (mod Λ ′ , mod Λ, mod Λ ′′ ) be a recollement. If dim mod Λ = 0, then dim mod Λ ′ = 0 = dim mod Λ ′′ ; that is, Λ is of finite representation type implies that so are Λ ′ and Λ ′′ ( [23]). Conversely, if dim mod Λ ′ = 0 = dim mod Λ ′′ , then dim mod Λ = 0 does not hold true in general.…”

Right Approximations and Recollements

“…The methods of constructing an A -resolution of an object in is similar to constructing a projective resolution in [8 [4 3) If the pair ( , ) er is an adjoint functor pair and the functor r is exact, then the left adjoint functor e preserves projective objects.…”

“…Now it plays an important role in algebraic geometry, representation theory, polynomial functor theory and ring theory, see for example [1]- [4] and references therein. Psaroudakis and Vitória observed that a recollement whose terms are module categories is equivalent to one induced by an idempotent element and is extensively studied (see [3], [4]). …”

Grade, dominant dimension and Gorenstein algebras

“…We will relate the grade of A/AeA as a left A-module to D(eA)-dominant dimension of A, where e is an idempotent of A and A/AeA is the quotient algebra of A modulo the idempotent ideal generated by e. Theorem 2.9. (See Psaroudakis [25,Theorem 3.10].) Let N be in A-mod and n be an integer.…”

“…The isomorphism A D(eA) ∼ = A Hom eAe (eA, D(eAe)) and [25,Proposition 3.4] imply that D(eA)-domdim M ≥ n + 1 if and only if Ext i A (X, M ) = 0 for any X ∈ A/AeA-mod and 0 ≤ i ≤ n. This implies that D(eA)-domdim A ≥ n + 1 if and only if grade A X ≥ n + 1 for any X ∈ A/AeA-mod. If grade A A/AeA = n, then Ext i A (X, A) = 0 for any X ∈ A/AeA-mod and 0 ≤ i ≤ n − 1 by [8, Lemma 2.2].…”