Relative rigid objects in triangulated categories

Let tau be a Krull-Schmidt, Horn-finite triangulated category with suspension functor [1]. Let R be a basic rigid object, Gamma the endomorphism algebra of R, and pr(R) subset of tau the sub-category of objects finitely presented by R. We investigate the relative rigid objects, i.e. R[1]-rigid objects of tau. Our main results show that the R[1]-rigid objects in pr(R) are in bijection with it tau-rigid Gamma-modules, and the maximal R[1]-rigid objects with respect to pr(R) are in bijection with support tau-tilting Gamma-modules. We also show that various previously known bijections involving support tau-tilting modules are recovered under respective assumptions. (C) 2018 Elsevier Inc. All rights reserved.