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.

中文翻译：

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万能扩展和支持τ倾斜模块

令R为A由幂等理想M的分裂扩展。我们给予的充分必要条件，以确保在支持τ -tilting [R -模诱导支持τ -tilting一个-module，相反，当支持τ -tilting一个-模诱导支持τ -tilting [R -模。我们研究A和R之间的同源关系。特别地，我们证明如果R是有限的全局维，则A具有有限的全球规模。作为一个应用，我们推导了一个τ倾斜模块的内同构代数在某些代数上的全局维的界。