Support τ-tilting modules and separating splitting tilting triples

Abstract

Adachi et al. have introduced support τ-tilting pairs from the viewpoint of mutation. Let (A, T, B) be a tilting triple. In this article, it is proved that there is a bijection between the left mutations of a basic support τ-tilting module T and the indecomposable splitting projective objects of , in the sense of Auslander and Smalø. If T is a separating and splitting tilting module, then there is a bijection between the basic τ-rigid pairs of A and of B. This bijection preserves the number of indecomposable direct summands of basic τ-rigid pairs, and hence restricts to a bijection between the basic support τ-tilting pairs of A and of B. Moreover, it restricts to a bijection between the basic partial tilting pairs of A and of B, and a bijection between the basic support tilting pairs of A and of B, in the sense of Ingalls and Thomas. This bijection also preserves mutations of support τ-tilting pairs, thus, the underlying graphs of the support τ-tilting quivers of A and of B are the same, but in general their orientations are not the same, and a sufficient and necessary condition so that left mutations are preserved is given.