Triads.
The present paper is a following up of the author's previous paper [\textit{W. Rump}, The category of lattices over a lattice finite ring, to appear in Algebr. Represent. Theory], where the author develops a theory of so-called \(L\)-functors, which is in some sense a functorial replacement of the Auslander-Reiten translation. In the paper under review the author is concerned with Krull-Schmidt categories with enough Ext-injective and enough Ext-projective objects, so that every split epimorphism has a kernel. The main result of the paper is that such a category has almost split sequences if and only if some category of short exact sequences is semilocal, cosemilocal and has enough semisimple objects. This in turn is equivalent to the fact that the category of two-term complexes with radical morphism up to homotopy has \(L\)-functors. On the way the author has to develop a theory that mimics the construction of a derived category in this more general setup. The author calls these structures triads and shows that they have interesting properties.
scientific article; zbMATH DE number 2136729
scientific article