Unbounded ladders induced by Gorenstein algebras

Abstract

Abstract. The derived category D(ModA) of a Gorenstein triangular matrix algebra A admits an unbounded ladder; and this ladder restricts toleft recollement of triangulated categories with Serre functors sits in a ladder of period 1; as an application, the singularity category of A admits a ladder of period 1. Recollements ([BBD]

2-recollements of singualrity categories and Gorenstein defect categories over triangular matrix algebras

“…Besides, the recollement structures over the triangular matrix algebra might be enriched in some suitable settings. For example, it was shown in [35] that if A, B and T are finite-dimensional Gorenstein algebras, then there exists a unbounded ladder of period 1 for the stable categories of Gorenstein projective modules (and hence for the singularity categories). Meanwhile, we note that the recollements under consideration in [27,28,34] are initially from the following 2-recollement of module categories:…”

“…While the diagram involving is called a ladder [1,6], if the recollement could be extended upwards and downwards. Generous evidences indicate that a recollement behaves better when it admits some extra adjoint functors, see [1,6,33,35] and references therein for instance.…”

Verdier quotients of homotopy categories

“…The first statement is the first row of [10,Theorem B]. The second statement can be shown using [24,Lemma 3.4(2)] and [10,Theorem B] for Z = X. For the convenience of the reader, we give a simple independent proof using the first statement.…”

Recollements, Cohen-Macaulay Auslander algebras and Gorenstein projective conjecture

“…n-recollements of unbounded derived categories of algebras unify the recollements of different kinds of bounded derived categories: a recollement of D − (mod) is a 2-recollement of D(Mod), a recollement of K b (proj) is a 3-recollement of D(Mod), and a recollement of D b (mod) is a 3-recollement of D(Mod) if the algebras are finite-dimensional over a field k. On the other hand, the language of n-recollements has proved powerful in clarifying the relationships between recollements of derived categories and certain homological properties of algebras [24]. We refer to [11,28] for more related topics on ladders and n-recollements.…”