Split-by-Nilpotent Extensions and Support τ-Tilting Modules

Abstract

Let R be a split extension of A by a nilpotent ideal M. We give necessary and sufficient conditions to ensure when a support τ-tilting R-module induces a support τ-tilting A-module and, conversely, when a support τ-tilting A-module induces a support τ-tilting R-module. We investigate homological relations between A and R. In particular, we prove that if R is of finite global dimension then A is of finite global dimension. As an application, we deduce a bound for the global dimension of the endomorphism algebra of a τ-tilting module over certain algebras.