Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures

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 \mathop{Ext}\nolimits ^i_\Lambda(Z,Z)=\mathop{Ext}\nolimits ^i_\Lambda(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.

We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over k=\mathbb F_2(q,H_1,H_2) with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.