Algebra · Result 207

The ℓ¹-Bass conjecture for all discrete groups

2 papersLean

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.

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The problem

What is being asked, where it comes from, and why it is hard. No proof needed.

▶Introduction to the problemin preparation

The solution proposed by OpenAI

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▶The proposed solutionin preparation

Papers 2

The ℓ¹-Bass Conjecture for Discrete Groups

OpenAI · October 5, 2026

We prove the ℓ¹-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ¹ group algebra is supported on finitely many conjugacy classes of finite-order elements.

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@misc{OAI:The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026,
  author = {{OpenAI}},
  title = {{The $\ell^1$-Bass Conjecture for Discrete Groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf}{OAI:The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026}},
  year = {2026}
}

The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

OpenAI · September 24, 2026

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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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}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

Scope

The formalization proves the complex Bass trace conjecture for every group: the Hattori–Stallings trace of each virtual class of finitely generated projective right \(\mathbb C[G]\)-modules vanishes on conjugacy classes of infinite-order elements. No finiteness, countability, or geometric hypothesis on \(G\) is imposed.

For torsion-free \(G\), it identifies the trace on \(K_0(\mathbb C[G])\) with the integral augmentation rank and proves the characteristic-zero Kaplansky idempotent conjecture: for every commutative unital domain \(R\) of characteristic zero, an idempotent in \(R[G]\) is \(0\) or \(1\).

Comparator links

Result	Comparator statement
Complex Bass trace vanishing and support	BassTrace.lean
Trace rank and characteristic-zero Kaplansky idempotents for torsion-free groups	BassTorsionFree.lean

Scope note on GitHub →