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

TL;DR

This paper investigates conditions under which Bongartz τ-complements of τ-rigid modules over a finite-dimensional algebra are preserved when extending the algebra via split nilpotent bimodules, providing necessary and sufficient criteria.

Contribution

It establishes necessary and sufficient conditions for Bongartz τ-complements to be preserved under split-by-nilpotent extensions of algebras.

Findings

Conditions for τ-rigidity preservation in module extensions

Criteria for Bongartz τ-complement transfer across extensions

Characterization of module behavior under split nilpotent extensions

Abstract

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.