Reduction of τ-Tilting Modules and Torsion Pairs

The class of support tau-tilting modules was introduced recently by Adachi et al. These modules complete the class of tilting modules from the point of view of mutations. Given a finite-dimensional algebra A, we study all basic support tau-tilting A-modules which have a given basic tau-rigid A-module as a direct summand. We show that there exist an algebra C such that there exists an order-preserving bijection between these modules and all basic support tau-tilting C-modules; we call this passage tau-tilting reduction. An important step in our proof is the formation of tau-perpendicular categories which are analogs of ordinary perpendicular categories. Finally, we show that tau-tilting reduction is compatible with silting reduction and 2-Calabi-Yau reduction in appropriate triangulated categories.