A Counterexample to Left-Right Symmetry for Diagonal Auslander-Regularity

We construct a finite-dimensional Auslander-regular algebra $R$ with global dimension three, which is right diagonal but not left diagonal; moreover, any simple right $R$-module with grade $i$ has projective dimension $i$, but there is a simple left $R$-module with grade one and projective dimension two. It answers a question raised by Iyama negatively.