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Extension conjecture for loops in the quiver

Prove that if the quiver Q of a bound quiver algebra Λ = kQ/I has a loop at a vertex u, then Ext^*_Λ({_u}k,{_u}k) is infinite; equivalently, Tor^Λ_*(k_u,{}_uk) is infinite.

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Background

The paper recalls the extension conjecture relating loops in the quiver to the infinitude of self-extensions of the corresponding simple module. The authors note that the conjecture is proved for monomial and special biserial algebras but remains open in general.

They use this conjecture to connect structural features of the quiver (presence of loops) with the infinite + and co+ global dimension conditions introduced earlier.

References

The extension conjecture is as follows, see . If there is a loop at a vertex $u$, then $Ext*({_u}k,{_u}k)$ is infinite - equivalently $Tor_*\Lambda(k_u, {_u}k)$ is infinite.

— Happel's question, Han's conjecture and $τ$-Hochschild (co)homology (2509.05135 - Cibils et al., 5 Sep 2025) in Subsection: Extension conjecture