Han’s conjecture on Hochschild homology and infinite global dimension

Prove that for every finite-dimensional associative algebra Λ over a field, if the global dimension of Λ is infinite, then the Hochschild homology HH_n(Λ) is nonzero for infinitely many n (i.e., the graded Hochschild homology is infinite).

Background

The authors recall Han’s conjecture, which predicts that infinite global dimension of Λ forces Hochschild homology to be nontrivial in infinitely many degrees. They also list classes of algebras for which the conjecture has been verified, indicating that the conjecture remains open in general.

This conjecture functions as a benchmark in the paper, where they introduce τ-Hochschild homology and connect it to a strengthened plus-type global dimension condition.

References

Y. Han conjectured in that if the global dimension of an algebra is infinite, then its Hochschild homology is infinite.

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