Han’s conjecture: finiteness of Hochschild homology implies smoothness

Determine whether, for every finite-dimensional algebra B over an algebraically closed field k, the condition that the Hochschild homology groups HH_n(B) are nonzero for only finitely many integers n implies that B is homologically smooth over k.

Background

Han’s conjecture asserts that if a finite-dimensional k-algebra has Hochschild homology concentrated in finitely many degrees, then it must be (homologically) smooth. Over perfect fields, smoothness is equivalent to finite global dimension, the condition under which the conjecture was originally formulated. Despite partial progress in special cases, the conjecture remains unresolved in general.

The present paper focuses on and refutes a differential graded (DG) generalization of Han’s conjecture by constructing a finite-dimensional DG algebra B with only finitely many nonzero Hochschild homology groups that is not smooth. However, this does not settle the original (non-DG) conjecture, which the authors note remains open.