有限性を仮定しない可換環論へのAI活用 (AI for Commutative Ring Theory without Finiteness Assumptions)

Abstract

I describe how ChatGPT was used in my research on non-Noetherian commutative ring theory. A counterexample showing that the Hamilton–Marley Cohen–Macaulay property is not preserved under localization took shape once the problem was reduced to constructing modules satisfying a few explicit conditions. ChatGPT then proposed constructions, which I verified by hand. By contrast, an attempt to analyze the Soublin–Alfonsi example, a coherent ring that is not stably coherent, remains inconclusive. Along the way, the model assigned to the ring the weak global dimension of its polynomial ring. Finally, I mention ongoing work (arXiv:2608.17879) on cohomological Cohen–Macaulayness, a local condition that implies local Cohen–Macaulayness in the sense of Hamilton–Marley.