量子完全交上的 Auslander--Reiten 猜想
The Auslander--Reiten conjecture for quantum complete intersections
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中文总结 AI 辅助
本文证明任意域上有限维量子完全交满足 Auslander--Reiten 猜想与 Tachikawa 第二猜想，通过刚性模分级扭转与自扩张判据实现。
中文摘要 AI 辅助
设 A 为任意域上的有限维量子完全交。我们证明：当有限维左 A-模 M 满足 Ext_A^1(M,M)=Ext_A^2(M,M)=0 时，M 必为投射模，且对交换参数的乘法阶数不作任何限制。因此，A 满足 Auslander--Reiten 猜想和 Tachikawa 第二猜想。证明结合了刚性模的可分级性、连续分级扭转以及 Avramov 和 Buchweitz 关于交换完全交的自扩张判据。我们推导出在分级扭转下普通自扩张群的比较公式，将所得判据应用于 Liu--Schulz 代数，并在四生成元情形下显式描述连续扭转。
英文摘要
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 \(\Ext_A^1(M,M)=\Ext_A^2(M,M)=0\), 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.
