algebra / Homological algebra
For a finite-dimensional algebra $A$, finite global dimension forces $\mathrm{HH}_n(A) = 0$ for all large $n$. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional $\mathbb{C}$-algebra with $\mathrm{HH}_n(A) = 0$ for every $n \geq 1$ and $\mathrm{gldim}\, A = \infty$, built by transporting Krah's phantom into a singularity category via one-periodic folding.
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Append-only history
The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.
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For a finite-dimensional algebra $A$, finite global dimension forces $\mathrm{HH}_n(A) = 0$ for all large $n$. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional $\mathbb{C}$-algebra with $\mathrm{HH}_n(A) = 0$ for every $n \geq 1$ and $\mathrm{gldim}\, A = \infty$, built by transporting Krah's phantom into a singularity category via one-periodic folding.
The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.
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