Han's Conjecture

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.