Don’t miss the “Note added in proof” just before the references: The normal form needs a little adjustment when the residue field is not algebraically closed.

Erratum: There is an error in the proof of the classification in dimensions higher than one. While the statement of the main theorem is correct, the argument in the paper does not cover the rings k[[x_0, ..., x_d]]/(x_d^2), where d > 1. Indeed, these rings do not have bounded CM type. Here is a proof of a more general result.

Don’t miss the “Note added in proof” just before the references: The normal form needs a little adjustment when the residue field is not algebraically closed. Also, there is an error in the proof of the classification in dimensions higher than one. While the statement of the main theorem is correct, the argument in the paper does not cover the rings k[[x_0, ..., x_d]]/(x_d^2), where d > 1. Indeed, these rings do not have bounded CM type. Here is a proof of a more general result.