Comparing $\tau$-tilting modules and $1$-tilting modules

Xiaojin Zhang; Xiao-Wu Chen; Zhibing Zhao; Zhi-Wei Li

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Comparing $\tau$-tilting modules and $1$-tilting modules

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arxiv 2501.02466 v1 pith:4QXNX7BO submitted 2025-01-05 math.RT math.RA

Xiao-Wu Chen , Zhi-Wei Li , Xiaojin Zhang , Zhibing Zhao This is my paper · ORCID

classification math.RTmath.RA

keywords tiltingmodulescharacterizeconditionconjecturedeloopingmodulesatisfying

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abstract

We characterize $\tau$-tilting modules as $1$-tilting modules over quotient algebras satisfying a tensor-vanishing condition, and characterize $1$-tilting modules as $\tau$-tilting modules satisfying a ${\rm Tor}^1$-vanishing condition. We use delooping levels to study \emph{Self-orthogonal $\tau$-tilting Conjecture}: any self-orthogonal $\tau$-tilting module is $1$-tilting. We confirm the conjecture when the endomorphism algebra of the module has finite global delooping level.

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