Abstract
In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and τ -tilting theory in module categories. Our main results show that relative rigid objects are in bijection with τ -rigid pairs, and also relative maximal rigid objects with support τ -tilting pairs under some assumptions. These results generalize the work by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. In addition, we introduce mutation of relative maximal rigid objects and show that any basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable complements. All results highlight new phenomena when they applied to exact categories.
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