Auslander–Reiten conjecture for quantum complete intersections

The conjecture asks whether, for a finite-dimensional module $M$ over the relevant algebra, vanishing of $\operatorname{Ext}^{1}(M,M)$ and $\operatorname{Ext}^{2}(M,M)$ forces $M$ to be projective. The new claim concerns all finite-dimensional quantum complete intersections over arbitrary fields.

Known results

• Quantum complete intersections with commutation parameters that are roots of unity were previously known cases (2020).

September 22, 2026 claimed proof

On September 22, 2026, the arXiv record for Weiheng Xia's preprint reported a proof without restrictions on parameter orders. It claims the Auslander–Reiten conjecture and Tachikawa's second conjecture for this class, including the Liu–Schulz family. The proof is unrefereed, so the resolution remains unverified.

Current status (as of September 2026): The conjecture is claimed for all finite-dimensional quantum complete intersections, but the claim is unrefereed and therefore unverified; the general Auslander–Reiten conjecture remains outside the result.