The Auslander-Reiten conjecture for Artin algebras

Claims Theorem 1.1: over $k=\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional triangular algebra $\Lambda$ with eight simple modules and $\mathrm{rad}(\Lambda)^4\ne0$, and a finite-dimensional nonprojective Gorenstein-projective module $Z$ with $\mathrm{Ext}^i(Z,Z)=\mathrm{Ext}^i(Z,\Lambda)=0$ for all $i\ge1$, persisting under every field extension (so also over an algebraically closed field of characteristic two). This disproves the Auslander-Reiten and Gorenstein-projective conjectures; $Z\oplus\Lambda$ also refutes the generator form. It does NOT touch the commutative Noetherian form of the conjecture, and gives no example in characteristic zero.

Claims Theorem 1.1: over $k=\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional triangular algebra $\Lambda$ with eight simple modules and $\mathrm{rad}(\Lambda)^4\ne0$, and a finite-dimensional nonprojective Gorenstein-projective module $Z$ with $\mathrm{Ext}^i(Z,Z)=\mathrm{Ext}^i(Z,\Lambda)=0$ for all $i\ge1$, persisting under every field extension (so also over an algebraically closed field of characteristic two). This disproves the Auslander-Reiten and Gorenstein-projective conjectures; $Z\oplus\Lambda$ also refutes the generator form. It does NOT touch the commutative Noetherian form of the conjecture, and gives no example in characteristic zero.