Two-term relative cluster tilting subcategories, T-tilting modules and silting subcategories

Let C be a triangulated category with shift functor [1] and Ra rigid subcategory of C. We introduce the notions of two-term R[1]-rigid subcategories, two-term (weak) R[1]-cluster tilting subcategories and two-term maximal R[1]-rigid subcategories. Our main result shows that there exists a bijection between the set of two-term R[1]-rigid subcategories of Cand the set of t-rigid subcategories of modR, which induces a one-to-one correspondence between the set of two-term weak R[1]-cluster tilting subcategories of Cand the set of support t-tilting subcategories of modR. This generalizes the main results in [15] where Ris a cluster tilting subcategory. When Ris a silting subcategory, we prove that the two-term weak R[1]-cluster tilting subcategories are precisely two-term silting subcategories in [9]. Thus the bijection above induces the bijection given by Iyama-Jorgensen-Yang in [9]. (C) 2020 Elsevier B.V. All rights reserved.