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

Abstract

Let C be a triangulated category with shift functor [1] and R a 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 C and the set of τ-rigid subcategories of modR, which induces a one-to-one correspondence between the set of two-term weak R[1]-cluster tilting subcategories of C and the set of support τ-tilting subcategories of modR. This generalizes the main results in [15] where R is a cluster tilting subcategory. When R is 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-Jørgensen-Yang in [9].