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.

摘要

足立等。从突变的角度引入了支持τ-倾斜对。让 ( A , T , B ) 是一个倾斜的三元组。在本文中，证明了基本支持τ-倾斜模块T的左突变与不可分解的分裂射影对象之间存在双射$\text{脸}吨$, 在 Auslander 和 Smalø 的意义上。如果T是分离和分裂倾斜模块，则A和B的基本τ -刚性对之间存在双射。该双射保留了基本τ -刚体对的不可分解直接被加数的数量，因此限制为A和B的基本支撑τ -倾斜对之间的双射。此外，它限制了的基本部分倾斜对之间的双射甲和乙，以及基本支承倾斜对之间的双射甲和乙，在英格尔斯和托马斯的意义上。这个双射还保留了支持τ倾斜对的突变，因此，A和B的支持τ倾斜颤动的底层图是相同的，但一般来说它们的方向是不一样的，是一个充分必要条件，所以给出了保留的左突变。