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The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
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Abstract
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.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Indira Chatterji, contribution to 100+ reactions to 100+ solutions (Proofs and Prompts, posted 2026-10-08). A co-author of the amenable-group case (Berrick–Chatterji–Mislin) writes: "I don't understand the proof, and I still don't believe that this conjecture could be true in general," and asks for a small group of colleagues to examine it.
- Johan Öinert, contribution to 100+ reactions to 100+ solutions (Proofs and Prompts, posted 2026-10-08). Öinert notes the paper "does not say a word about the characteristic p case" and argues that the release's claimed characteristic-2 counterexample to Kaplansky's direct finiteness conjecture would also give one to the idempotent conjecture in characteristic 2; he also criticizes the write-up style.
- Roman Sauer, contribution to 100+ reactions to 100+ solutions (Proofs and Prompts, posted 2026-10-08). Sauer writes that a trick replacing a manifold by a measurable foliated object, from his work with Braun and Alpert's extension of it, "now appear in OpenAI's deeply creative solutions of problems 207 and 335"; both works are cited in this paper's geometric part. [family-level]
Formalization
- OpenAI / mathcommunity mirror, Lean scope note 207 (release Lean docs, 2026-10-06). The note says the formalization proves the complex Bass trace theorem for all groups and, for torsion-free groups, the trace/augmentation-rank identity and the characteristic-zero idempotent statement. [family-level]
- OpenAI, BassTrace.lean (openai/math Comparator challenge). The Lean statement
bassTraceModules_vanishing_and_support: the Hattori–Stallings coefficient vanishes at non-finite-order classes and is supported on finite-order classes (Theorem 1.1). - OpenAI, BassTorsionFree.lean (openai/math Comparator challenge). The Lean statement
torsion_free_corollary, covering both parts of Corollary 1.2 (trace equals augmentation rank, and idempotents in R[G] are 0 or 1 for characteristic-zero domains R).
Possibly relevant literature (found by the agent, not checked)
- Chatterji, Mislin, On the Idempotent Conjecture for Sidki Doubles (arXiv:2204.02176, 2022). A partial result on the idempotent conjecture for complex group rings of Sidki doubles of torsion-free groups, a special case of the setting of Corollary 1.2(ii).
- Klein, Naef, Poincaré Complex Diagonals and the Bass trace Conjecture (arXiv:2208.06289; Proc. Roy. Soc. Edinburgh A 154 (2024)). Relates Poincaré embeddings of the diagonal to the Reidemeister trace in connection with the Bass trace conjecture, relevant to the paper's corollary on homotopy idempotents and fixed points.
- Ronghui Ji and Bobby Ramsey, Cyclic Cohomology for Discrete Groups and its Applications (author-hosted survey, undated). Section 3.3 discusses cyclic-homological approaches to Bass trace vanishing and ℓ¹ analogues, background for the claims about support on finite-order conjugacy classes. [family-level]
Digests and expositions
- Open Mathematics Courses / Claude Opus 5.5, The Bass trace conjecture for group rings (Open Mathematics Courses, undated). A nine-lesson exposition following the manuscript's path sums, Pfaffian cocycle, separating filtrations, and idempotent consequences; it states it is AI-written and self-checked, with independent review incomplete. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 207
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026,
author = {{OpenAI}},
title = {{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture}},
howpublished = {OpenAI Math Release preprint
\href{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}{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/207.md. The Lean statements have not been checked by people against the paper's claims.
Family 207: The ℓ¹-Bass conjecture for all discrete groups
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $\ell^1(G)$ are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.
Other papers in this family:
- The ℓ¹-Bass Conjecture for Discrete Groups · October 5, 2026