We generalize Ringel and Schmidmeier's theory on the Auslander-Reiten translation of the submodule category S 2 .A/ to the monomorphism category S n .A/; the category consists of all chains of .n 1/ composable monomorphisms of A-modules. As in the case of n D 2, S n .A/ has Auslander-Reiten sequences, and the Auslander-Reiten translation S of S n .A/ can be explicitly formulated via of A-mod. Furthermore, if A is a selfinjective algebra, we study the periodicity of S on the objects of S n .A/ and of the Serre functor F S on the objects of the stable monomorphism category S n .A/. In particular, 2m.nC1/ S X Š X for X 2 S n .ƒ.m; t //, and F m.nC1/ S X Š X for X 2 S n .ƒ.m; t //, where ƒ.m; t/, m 1, t 2, are the selfinjective Nakayama algebras.
It is proved that strong exact Borel subalgebras of quasi-hereditary algebra are conjugate to each other and are in bijection with the collection of maximal semisimple subalgebras.
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]
For an acyclic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q, A) and prove that Mon(Q, A) has enough injective objects.
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