{"id":"openai-math/The-Bass-trace-conjecture-for-complex-group-rings","pdf":"https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026","tags":["group-rings","idempotents","algebraic-k-theory"],"score":0,"title":"The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture","domain":"Algebra","family":"207","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.RA","summary":"We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in $`RG`$ are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":2,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=This%20trick%20and%20these%20two%20papers%20now%20appear%20in%20OpenAI%E2%80%99s%20deeply%20creative%20solutions%20of%20problems%20207%20and%20335","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"This trick and these two papers now appear in OpenAI’s deeply creative solutions of problems 207 and 335 – problems so different, they could not be farther apart! I am stunned. What do I take away from this night? We cannot defensively narrow our focus to what we currently regard as human…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"},{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=One%20of%20the%20papers%20is%20a%20solution%20to%20the%20Bass%20trace%20conjecture","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"One of the papers is a solution to the Bass trace conjecture, a beautiful conjecture made by Hyman Bass in 1976, implying the older idempotent conjecture (from the 50ies maybe?) that there should be non non-trivial idempotent in the group ring of a torsion-free group. I was privileged enough to…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"}],"formalized":true,"family_title":"The ℓ¹-Bass conjecture for all discrete groups"}