On a conjecture of Auslander and Reiten

In studying Nakayamaʹs 1958 conjecture on rings of infinite dominant dimension, Auslander and Reiten proposed the following generalization: Let Λ be an Artin algebra and M a Λ-generator such that ExtiΛ(M,M)=0 for all i 1; then M is projective. This conjecture makes sense for any ring. We establish Auslander and Reitenʹs conjecture for excellent Cohen–Macaulay normal domains containing the rational numbers, and slightly more generally.