A counterexample to DG version of Han's conjecture

Abstract

In 2004, Han proposed the following conjecture: let $B$ be a finite-dimensional $k$-algebra. If ${\mathrm{HH}}_n(B)\ne 0$ for only finitely many $n\in \mathbb{Z}$, then $B$ is smooth. This conjecture can be generalized to the DG setting: let $B$ be a finite-dimensional DG $k$-algebra. If ${\mathrm{HH}}_n(B)\ne 0$ for only finitely many $n\in \mathbb{Z}$, then $B$ is smooth. In this note, we show that the DG generalization of Han's conjecture is false.