Let (F , Σn, ) be an (n+2)-angulated Krull-Schmidt category and A ⊆ F an n-extension closed, additive and full subcategory with Hom F (ΣnA , A ) = 0. Then A naturally carries the structure (A , E A ) of an n-exact category in the sense of [Jas16, definition 4.2], arising from short (n+2)-angles in (F , Σn, ) with objects in A and there is a binatural and bilinear isomorphism YExt n (A ,E A ) (An+1, A0) ∼ = Hom F (An+1, ΣnA0) for A0, An+1 ∈ A . For n = 1 this has been shown in [Dye05] and we generalize this result to the case n > 1. On the journey to this result, we also develop a technique for harvesting information from the higher octahedral axiom (N4*) as defined in [BT13, section 4]. Additionally, we show that the axiom (F3) for pre-(n + 2)-angulated categories, stating that a commutative square can be extended to a morphism of (n + 2)-angles and defined in [GKO13, definition 2.1], implies a stronger version of itself.
Suppose (C, E, s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C, E, s) to (resp. weakly) idempotent complete n-exangulated categories. We note that our methods of proof differ substantially from the extriangulated and (n + 2)-angulated cases. However, our constructions recover the known structures in the established cases up to nexangulated isomorphism of n-exangulated categories.Acknowledgements. The authors would like to thank Theo Bühler, Ruben Henrard and Adam-Christiaan van Roosmalen for useful email communications, and Andrew Brooke-Taylor and Peter Jørgensen for helpful discussions.
We show how to obtain a minimal projective resolution of finitely generated modules over an idempotent subring Γe := (1−e)R(1−e) of a semiperfect noetherian basic ring R by a construction inside modR. This is then applied to investigate homological properties of idempotent subrings Γe under the assumption of R/ 1 − e being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module Se := eR/ rad eR with Ext 1 R (Se, Se) = 0 is self-orthogonal, that is Ext k R (Se, Se) vanishes for all k ≥ 1, whenever gldim R and pdim eR(1 − e) Γe are finite (see [IP16, conjecture 4.13]). Indeed a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose e ∈ R is an idempotent such that all idempotent subrings Γ sandwiched between Γe and R, that is Γe ⊆ Γ ⊆ R, have finite global dimension. Then the simple summands of Se can be numbered S1, . . . , Sn such that Ext k R (Si, Sj) = 0 for 1 ≤ j ≤ i ≤ n and all k > 0.
For a hereditary, finite dimensional k-algebra A the Coxeter transformation is a linear endomorphism ΦA : K0(modA) → K0(modA) of the Grothendieck group K0(modA) of modA such that ΦA[M ] = [τAM ] for every non-projective indecomposable A-module M , that is the Auslander-Reiten translation τA induces a linear endomorphism of K0(modA). It is natural to ask whether there are other algebras A having a linear endomorphism ΦA ∈ End Z (K0(modA)) with ΦA[M ] = [τAM ] for all non-projective M ∈ indA. We will show that this is the case for all Nakayama algebras. Conversely, we will show that if an algebra A = kQ/I, where Q is a connected and non-acyclic quiver and I kQ is an admissible ideal, admits such a linear endomorphism then A is already a cyclic Nakayama algebra.
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