Relative rigid objects in extriangulated categories (Q2065612)

The authors study the relation between relative cluster tilting theory in extriangulated categories and \(\tau\)-tilting theory in module categories, generalizing results of Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. First, they study properties of \(R\)-rigid objects over a Krull-Schmidt, Hom-finite, extriangulated category. Second, gives a bijection between certain isomorphisms classes of basic \(R\)-rigid objects and classes of basic \(\tau\)-rigid pairs of modules (Thm 3.13); which allows to give an equivalent characterization on tilting modules (Thm 3.14). Third, study the relationship between \(R\)-rigid, rigid and d-rigid (Thm 3.17). Finally, with an example, they illustrated the main results.