Artin–Hochster conjecture for commuting varieties

Prove or disprove that, for the classical commuting variety of pairs of n × n matrices over an algebraically closed field, the coordinate ring defined by the entries of AB − BA is reduced and Cohen–Macaulay for every n.

Background

The paper contrasts its completely solved iterated-commutator problem for 2 × 2 matrices with the classical commuting variety of pairs of n × n matrices. The cited Artin–Hochster conjecture asks for reducedness and Cohen–Macaulayness of the corresponding coordinate ring in all matrix dimensions. The paper further records that Cohen–Macaulayness remains open for n ≥ 5 and reducedness remains open for n ≥ 4.

References

Whether the coordinate ring $\mathbb{F}\lbrack A,B\rbrack/I$, $I$ the ideal generated by the entries of $AB - BA$, is reduced and Cohen-Macaulay for every $n$ is a much harder question, attributed to Michael Artin and Melvin Hochster -- an unpublished, informally-circulated conjecture with no single dateable original source , stated precisely as ``Conjecture 1.1'' in Majidi-Zolbanin and Snapp .

— The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices  (2609.05386 - Snellman, 4 Sep 2026) in Section 1, Introduction