Projective dimension and commuting variety of a reductive Lie algebra

Jean-Yves Charbonnel

Published 2020-06-23Version 1

The commuting variety of a reductive Lie algebra g is the underlying variety of a well defined subscheme of g x g. In this note, it is proved that this scheme is normal and Cohen-Macaulay. In particular, its ideal of definition is a prime ideal. As a matter of fact, this theorem results from a so called Property (P) for a simple Lie algebra. This property says that some cohomology complexes are exact.