Self-orthogonal tau-tilting conjecture

Prove that every self-orthogonal -tilting module over an Artin algebra is a tilting module, equivalently that self-orthogonality forces faithfulness for -tilting modules.

Background

The paper studies when a self-orthogonal -tilting module over an Artin algebra must be faithful. Faithfulness is equivalent to the module being a 1-tilting module, so the unresolved conjecture asks whether self-orthogonality alone suffices to promote every -tilting module to a tilting module.

The paper proves the conjecture for several classes, including radical square zero Artin algebras and semisimple-source triangular matrix algebras with local terminal blocks, but explicitly states that the general case remains unresolved.

References

In fact, we conjectured that a self-orthogonal $\tau$-tilting module is a tilting module , which is called the self-orthogonal $\tau$-tilting conjecture in . Moreover, it is shown that the conjecture holds for algebras of finite global dimension , gentle algebras , Gorenstein CM-finite algebras , algebras of finite representation type and minimal representation infinite algebras . In general, the conjecture is open.

— Self-Orthogonal $τ$-Tilting Modules and Tilting Modules II: Annihilator Separation  (2608.27937 - Zhang, 28 Aug 2026) in Section 1, Introduction

Can a nonzero nilpotent ideal and a $\tau$-tilting module satisfy all conditions in Proposition \ref{prop:counterexample-profile} simultaneously? A negative answer in a stable class of algebras would give a new route from self-orthogonality to tilting.

— Self-Orthogonal $τ$-Tilting Modules and Tilting Modules II: Annihilator Separation  (2608.27937 - Zhang, 28 Aug 2026) in Section 5, final two Question environments