Almost split morphisms in subcategories of triangulated categories

For a suitable triangulated category [Formula: see text] with a Serre functor [Formula: see text] and a full precovering subcategory [Formula: see text] closed under summands and extensions, an indecomposable object [Formula: see text] in [Formula: see text] is called Ext-projective if Ext[Formula: see text]. Then there is no Auslander–Reiten triangle in [Formula: see text] with end term [Formula: see text]. In this paper, we show that if, for such an object [Formula: see text], there is a minimal right almost split morphism [Formula: see text] in [Formula: see text], then [Formula: see text] appears in something very similar to an Auslander–Reiten triangle in [Formula: see text]: an essentially unique triangle in [Formula: see text] of the form [Formula: see text] where [Formula: see text] is an indecomposable not in [Formula: see text] and [Formula: see text] is a [Formula: see text]-envelope of [Formula: see text]. Moreover, under some extra assumptions, we show that removing [Formula: see text] from [Formula: see text] and replacing it with [Formula: see text] produces a new subcategory of [Formula: see text] closed under extensions. We prove that this process coincides with the classic mutation of [Formula: see text] with respect to the rigid subcategory of [Formula: see text] generated by all the indecomposable Ext-projectives in [Formula: see text] apart from [Formula: see text]. When [Formula: see text] is the cluster category of Dynkin type [Formula: see text] and [Formula: see text] has the above properties, we give a full description of the triangles in [Formula: see text] of the form [Formula: see text] and show under which circumstances replacing [Formula: see text] by [Formula: see text] gives a new extension closed subcategory.

Relative n-rigid objects in (n + 2)-angulated categories

Let [Formula: see text] be an algebraically closed field, [Formula: see text] an integer, [Formula: see text] a [Formula: see text]-linear Hom-finite [Formula: see text]-angulated category with [Formula: see text]-suspension functor [Formula: see text], a Serre functor [Formula: see text], and split idempotents. Let [Formula: see text] be a basic [Formula: see text]-rigid object and [Formula: see text] the endomorphism algebra of [Formula: see text]. We introduce the notion of relative [Formula: see text]-rigid objects, i.e. [Formula: see text]-rigid objects of [Formula: see text]. Then we show that the basic maximal [Formula: see text]-rigid objects in [Formula: see text] are in bijection with basic maximal [Formula: see text]-rigid pairs of [Formula: see text]-modules when every indecomposable object in [Formula: see text] is [Formula: see text]-rigid. As an application, we recover a result in Jacobsen–Jørgensen [Maximal [Formula: see text]-rigid pairs, J. Algebra 546 (2020) 119–134].

The generalized Auslander–Reiten duality on an exact category

We introduce a notion of generalized Auslander–Reiten duality on a Hom-finite Krull–Schmidt exact category [Formula: see text]. This duality induces the generalized Auslander–Reiten translation functors [Formula: see text] and [Formula: see text]. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text]. A non-projective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the first term of an almost split conflation. We study the generalized Auslander–Reiten duality on the category of finitely presented representations of locally finite interval-finite quivers.