Han's conjecture on Hochschild homology and smoothness

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Let BB be a finite-dimensional algebra over an algebraically closed field kk. Write HH⁡n(B)\operatorname{HH}_n(B) for the Hochschild homology groups of BB. The algebra BB is (homologically) smooth over kk when it is perfect as a bimodule over its enveloping algebra. Han's conjecture. If HH⁡n(B)\operatorname{HH}_n(B) is nonzero for only finitely many n∈Zn\in\mathbb{Z}, then BB is (homologically) smooth over kk. The conjecture is a classical statement for finite-dimensional algebras; this paper's abstract distinguishes it from the DG generalization, which the paper disproves. The supplied text does not state whether Han's original conjecture has been resolved.

References

Primary source

Yeqin Liu and Yu Shen, “A counterexample to DG version of Han's conjecture”, arXiv:2512.12460 (2025).

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