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.