Reduction of $\tau$-tilting modules and torsion pairs

The class of support $\tau$-tilting modules was introduced recently by Adachi, Iyama and Reiten. 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 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 process $\tau$-tilting reduction. An important step in this process 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 appropiate triangulated categories.