For every finite-dimensional associative algebra A over a field K, the quotient of its Hochschild cohomology ring by the ideal of nilpotent elements is a finitely generated K-algebra; equivalently, nobreak operatornameHH^(A)/G(N) is finitely generated, where N denotes the nilpotent elements and G(N) the ideal they generate.

Snashall–Solberg conjecture

The Snashall–Solberg conjecture proposed finite generation for Hochschild cohomology of finite-dimensional algebras. Its original form is already known to fail, and a new paper claims a counterexample that is both selfinjective and symmetric.

Known results

In characteristic $2$ , the one-point extension $B=\Gamma[M]$ , with $\Gamma=K(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ and $M=K\mathbb{Z}_{2}$ , was identified as an initial counterexample to the original finite-generation conjecture (Psaroudakis, 2016).
The refined question of whether $\operatorname{HH}^{*}(A)/\mathcal{G}(\mathcal{N})$ is finitely generated was left open (Psaroudakis, 2016).

August 2026 symmetric counterexample

Edward L. Green and S. O. Ivanov claim a selfinjective, symmetric counterexample, extending the failure beyond nonsymmetric algebras. The claim is currently unrefereed and has no independent verification in the retrieved sources.

Current status (as of August 2026): The original conjecture is contradicted by a known nonsymmetric example, while the claimed selfinjective symmetric counterexample remains unverified and the refined quotient-finiteness question remains open.

Snashall–Solberg conjecture

Progress summary

Primary source

Additional references

Solutions 0