On support τ-tilting modules over endomorphism algebras of rigid objects

We consider a Krull–Schmidt, Hom-finite, 2-Calabi–Yau triangulated category with a basic rigid object T, and show a bijection between the set of isomorphism classes of basic rigid objects in the finite presented category pr T of T and the set of isomorphism classes of basic τ-rigid pairs in the module category of the endomorphism algebra End_c(T)_op. As a consequence, basic maximal objects in pr T are one-to-one correspondence to basic support τ-tilting modules over End_c(T)_op. This is a generalization of correspondences established by Adachi–Iyama–Reiten.