{
  "id": "bass-trace-conjecture",
  "title": "The Bass trace conjecture for group rings",
  "description": "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.",
  "units": [
    {
      "id": "traces-of-projective-modules",
      "title": "Traces of projective modules",
      "reader": "traces-of-projective-modules.html",
      "source": "src/traces-of-projective-modules.md",
      "source_sha256": "5de75e9a459392527ddb9deae5bf5cb416a48f01cf7051ca3ea36d0b840f753c",
      "status": "author_self_checked"
    },
    {
      "id": "path-sums-of-an-idempotent",
      "title": "Path sums of an idempotent",
      "reader": "path-sums-of-an-idempotent.html",
      "source": "src/path-sums-of-an-idempotent.md",
      "source_sha256": "33b91f93d4a2853fe0171bc4557b204d937b341a3a0b996af256ac992324c09c",
      "status": "author_self_checked"
    },
    {
      "id": "exterior-chains-and-sparse-cycles",
      "title": "Exterior chains and sparse cycles",
      "reader": "exterior-chains-and-sparse-cycles.html",
      "source": "src/exterior-chains-and-sparse-cycles.md",
      "source_sha256": "9dbcab67093bbd787c09e6c53280b83ed5ef96eaa35d3d184fe944052f58fd6a",
      "status": "author_self_checked"
    },
    {
      "id": "a-pfaffian-cocycle",
      "title": "A Pfaffian cocycle",
      "reader": "a-pfaffian-cocycle.html",
      "source": "src/a-pfaffian-cocycle.md",
      "source_sha256": "ff320515dfd0e74b4215dda8862abbc940620d46519eb6b6e5441c8b42d24c83",
      "status": "author_self_checked"
    },
    {
      "id": "a-transport-metric-with-small-balls",
      "title": "A transport metric with small balls",
      "reader": "a-transport-metric-with-small-balls.html",
      "source": "src/a-transport-metric-with-small-balls.md",
      "source_sha256": "97fd16043713dd4fbd3f45f2d1afb4fa07437744fede9cc409f2d895816b5b66",
      "status": "author_self_checked"
    },
    {
      "id": "separating-filtrations",
      "title": "Separating filtrations",
      "reader": "separating-filtrations.html",
      "source": "src/separating-filtrations.md",
      "source_sha256": "14a9f1dc48814c5823c3dd0025f8dbdb6318f0033e27aecf5eb8d9c9b917c46f",
      "status": "author_self_checked"
    },
    {
      "id": "near-minimal-separators",
      "title": "Near-minimal separators",
      "reader": "near-minimal-separators.html",
      "source": "src/near-minimal-separators.md",
      "source_sha256": "1a486361559b42c9bed4765fd68385dcc071b0bd9228b8c7294148cd3666e08c",
      "status": "author_self_checked"
    },
    {
      "id": "from-filtrations-to-chains",
      "title": "From filtrations to chains",
      "reader": "from-filtrations-to-chains.html",
      "source": "src/from-filtrations-to-chains.md",
      "source_sha256": "9d53eafd0c519a7ecf350ffbaf516dfb9e6e46e502a180843c054464ba495b8c",
      "status": "author_self_checked"
    },
    {
      "id": "idempotents-and-integral-traces",
      "title": "Idempotents and integral traces",
      "reader": "idempotents-and-integral-traces.html",
      "source": "src/idempotents-and-integral-traces.md",
      "source_sha256": "611394dea2b8c59124b6d8b15ae5eb676f1aedf341241e90cde29747663a88bf",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "programme lesson The regular trace and spectral phase (course the-kervaire-conjecture), Lemma 1.3: the trace on matrices over the group von Neumann algebra",
    "programme lesson Directly finite group algebras (course kaplansky-direct-finiteness), Lemma 3.2 (range projections) and Lemma 3.3 (finitely generated fields of characteristic zero embed in C)",
    "core course Real Analysis II (Lebl, Basic Analysis II): Proposition 8.2.4, Theorem 7.4.11, Theorem 10.2.3, Propositions 10.5.1, 10.5.5, 10.5.7, Theorem 10.7.2"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}