Bongartz $\tau$-Complements Over Split-By-Nilpotent Extensions

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E, and let M be a $\tau$-rigid C-module with U its Bongartz $\tau$-complement. If the induced module, $M{\otimes }_{C}B$, is $\tau$-rigid as a B-module, we give a necessary and sufficient condition for $U{\otimes }_{C}B$ to be its Bongartz $\tau$-complement in mod B. If M is $\tau$-rigid in mod B, we again provide a necessary and sufficient condition for $U{\otimes }_{C}B$ to be its Bongartz $\tau$-complement in mod B.