Two classes of algebras with infinite Hochschild homology

We prove without any assumption on the ground field that higher Hochschild homology groups do not vanish for two large classes of algebras whose global dimension in not finite.

💡 Research Summary

The paper investigates the relationship between global dimension and Hochschild homology for finite‑dimensional algebras over an arbitrary ground field. Building on Han’s conjecture—which predicts that a finite‑dimensional algebra whose Hochschild homology vanishes in sufficiently high degrees must have finite global dimension—the author identifies two broad families of algebras for which all higher Hochschild homology groups are non‑zero, thereby providing strong evidence for the “infinite global dimension ⇒ non‑vanishing high‑degree Hochschild homology” direction of the conjecture.
The first family consists of path algebras kQ/I whose underlying quiver Q contains at least one oriented cycle and whose defining ideal I does not kill all powers of that cycle. By constructing the bar resolution explicitly, the author shows that for each integer n≥1 the class of the n‑fold concatenation of the oriented cycle yields a non‑trivial Hochschild n‑cycle. The boundary map in the bar complex never annihilates these cycles because the relations in I preserve the cyclic structure, so HHₙ(A)≠0 for every n. This argument works without any restriction on the characteristic of the base field.
The second family is formed by tensor products of a radical‑square‑zero algebra R (i.e., rad(R)²=0) with a truncated polynomial algebra S=k