The Bass trace conjecture for group rings
A finitely generated projective module over a group ring has a trace with one coefficient for each conjugacy class. Bass conjectured that over the integers only the identity class carries a coefficient, and over the complex numbers that every class of elements of infinite order has coefficient zero. OpenAI proved both in 2026, for every group. For torsion-free groups this gives Kaplansky's idempotent conjecture in characteristic zero: the group ring has no idempotents other than 0 and 1. The proof turns an idempotent into simplicial cycles of every even dimension, detects its trace coefficient with a Pfaffian cocycle, and shows by a separating-set argument that such sparse cycles carry no invariant cohomology in high dimension. The course is an exposition of OpenAI's proof (2026); it also proves Linnell's restriction for integral group rings by an elementary congruence for traces of p-th powers.
Nine lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Traces of projective modules
- Path sums of an idempotent
- Exterior chains and sparse cycles
- A Pfaffian cocycle
- A transport metric with small balls
- Separating filtrations
- Near-minimal separators
- From filtrations to chains
- Idempotents and integral traces
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Cohomological and geometric dimension (A finitely generated counterexample to the Eilenberg-Ganea conjecture)
- Traces of projections and the Baum–Connes conjecture (A projection of irrational trace and the Baum–Connes conjecture)
- The Kadison–Kaplansky conjecture and scalar compression (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
- Directly finite group algebras (Kaplansky's direct-finiteness conjecture)
- The regular trace and spectral phase (The Kervaire conjecture)
- Group algebras and zero divisors (A torsion-free group algebra with zero divisors)
- Groups, rings, modules and group rings (recalled in the first lesson).
- Linear algebra: determinants, exterior products are recalled (core course Linear Algebra).
- Multivariable integration: Fubini's theorem and volumes of polytopes (core course Real Analysis II).