Let X and Y be square generic matrices of indeterminates. The commutator matrix of X and Y is defined to be C=XY-YX. Hochster conjectured that the vanishing of every entry of C defines a domain in 1984, which is still open for n>3. Since then researchers have studied ideals related to commutator matrices, usually defined by the vanishing of some entries of C. In this talk, I will prove a recent theorem that ideals defined by the vanishing of the cross-diagonal (diagonal + anti-diagonal) of commutator matrices define F-regular rings.
