Auslander-Reiten (d+2)-angles in subcategories and a (d+2)-angulated generalisation of a theorem by Bruning

Let Phi be a finite dimensional algebra over an algebraically closed field k and assume gldim Phi <= d, for some fixed positive integer d. For d = 1, Bruning proved that there is a bijection between the wide subcategories of the abelian category mod Phi and those of the triangulated category D-b (mod Phi). Moreover, for a suitable triangulated category M, Jorgensen gave a description of Auslander-Reiten triangles in the extension closed subcategories of M. In this paper, we generalise these results for d-abelian and (d + 2)-angulated categories, where kernels and cokernels are replaced by complexes of d + 1 objects and triangles are replaced by complexes of d + 2 objects. The categories are obtained as follows: if F subset of mod Phi is a d-cluster tilting subcategory, consider (F) over bar := add{Sigma(id) F vertical bar i is an element of Z} subset of D-b(mod Phi). Then.F is d-abelian and plays the role of a higher mod Phi having for higher derived category the (d + 2)-angulated category (F) over bar. Crown Copyright (C) 2018 Published by Elsevier B.V. All rights reserved.