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 $\text{Fac}T$, 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.

Keywords:

• Left mutation
• separating and splitting tilting triple
• τ-rigid
• splitting projective object
• splitting torsion pair
• support τ-tilting pair
• support tilting pair
• support τ-tilting quiver

2020 Mathematics Subject Classification:

• 16G10
• 16G20
• 16G70
• 18E40
• 18E35

Acknowledgement

We thank the referee for helpful comments.

Additional information

Funding

Supported by the NNSFC (National Natural Science Foundation of China) Grant No. 11971304.