Comparing $τ$-tilting modules and $1$-tilting modules
Xiao-Wu Chen, Zhi-Wei Li, Xiaojin Zhang, Zhibing Zhao
TL;DR
This work investigates the relationship between $τ$-tilting and $1$-tilting modules by establishing two new characterizations through annihilators and quotient algebras, using tensor-vanishing and Tor-vanishing conditions. It introduces the delooping level as a tool to study the Self-orthogonal $τ$-tilting Conjecture, proving that a self-orthogonal $τ$-tilting module is $1$-tilting under hypotheses on the delooping level of its endomorphism algebra. The paper also develops a framework of homological conjectures (SWC, SFC, S$τ$C) and provides several known cases where these conjectures hold, along with new criteria connecting those conjectures to delooping levels. The results give concrete, elementary criteria to transfer properties between $τ$-tilting and $1$-tilting modules and offer a pathway to verify long-standing conjectures in broader classes of algebras. Overall, the work deepens the connections between tilting theories and homological dimensions via novel invariants and vanishing conditions with potential applications in representation theory and beyond.
Abstract
We characterize $τ$-tilting modules as $1$-tilting modules over quotient algebras satisfying a tensor-vanishing condition, and characterize $1$-tilting modules as $τ$-tilting modules satisfying a ${\rm Tor}^1$-vanishing condition. We use delooping levels to study \emph{Self-orthogonal $τ$-tilting Conjecture}: any self-orthogonal $τ$-tilting module is $1$-tilting. We confirm the conjecture when the endomorphism algebra of the module has finite global delooping level.