The Auslander--Reiten conjecture for quantum complete intersections

Let $A$ be a finite-dimensional quantum complete intersection over an arbitrary field. We prove that a finite-dimensional left $A$-module $M$ is projective whenever ${\text{Ext}}_A^1(M,M)={\text{Ext}}_A^2(M,M)=0$, with no restriction on the multiplicative orders of the commutation parameters. Consequently, $A$ satisfies the Auslander–Reiten conjecture and Tachikawa’s second conjecture. The proof combines the gradability of rigid modules with successive graded twists and the self-extension criterion of Avramov and Buchweitz for commutative complete intersections. We derive a formula comparing ordinary self-extension groups under graded twisting, apply the resulting criterion to the Liu–Schulz algebras, and describe the successive twists explicitly in the four-generator case.