A counterexample to DG version of Han's conjecture

TL;DR

This paper provides a counterexample demonstrating that the DG (differential graded) generalization of Han's conjecture, which relates finite Hochschild homology to smoothness of algebras, does not hold.

Contribution

The paper constructs a specific counterexample showing that the DG version of Han's conjecture is false, challenging assumptions about the relationship between Hochschild homology and smoothness.

Findings

Counterexample disproves the DG Han's conjecture

Finite Hochschild homology does not imply smoothness in DG algebras

Challenges previous generalizations of Han's conjecture

Abstract

In 2004, Han proposed the following conjecture: let $B$ be a finite-dimensional $k$-algebra. If $\mathrm{HH}_{n}(B)\neq 0$ 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 $\mathrm{HH}_{n}(B)\neq 0$ 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.