Reduction of τ-tilting modules and torsion pairs

The class of support τ -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 τ -tilting A-modules which have a given basic τ -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 τ -tilting C-modules; we call this passage τ -tilting reduction. An important step in our proof is the formation of τ -perpendicular categories which are analogs of ordinary perpendicular categories. Finally, we show that τ -tilting reduction is compatible with silting reduction and 2-Calabi–Yau reduction in appropriate triangulated categories.