An explicit counterexample to the Auslander-Reiten conjecture

We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over $k={\mathbb{F}}_2(q,H_1,H_2)$ and a finite-dimensional nonprojective left module Z such that ${\mathrm{Ext}}_{\Lambda }^i(Z,Z)={\mathrm{Ext}}_{\Lambda }^i(Z,\Lambda )=0$ for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.