τ-tilting modules, depth and delooping level · arXivDesk2606.11684Jun 10, 202613 pages, generalized to support tau-tilting modules. comments are welcome
τ-tilting modules, depth and delooping level
Mingfei Xu, Xiaojin Zhang
Abstract
Let A be a finite-dimensional basic algebra over an algebraically closed field K, T a finitely generated support τ-tilting right A-module and B=EndAT
. Denote by
the subcategory of finitely generated right
-modules generated by
. We define the depth relative to
and the delooping level relative to
and show that the finitistic dimension of the opposite algebra of
is bounded by the depth of
relative to
and the delooping level of
relative to
. We give applications to the finitistic dimension conjecture. More precisely, we show that if
is a minimal representation infinite algebra or an algebra of finite representation type, then the finitistic dimension of
is finite.