Algebra · Result 203

Donovan's conjecture over fields and complete mixed-characteristic DVRs

2 papers

Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Donovan's Conjecture over Algebraically Closed Fields

OpenAI · September 24, 2026

We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.

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Cite

@misc{OAI:Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026,
  author = {{OpenAI}},
  title = {{Donovan's Conjecture over Algebraically Closed Fields}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf}{OAI:Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026}},
  year = {2026}
}

Integral Donovan Finiteness over Witt Vectors

OpenAI · September 25, 2026

We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over \(W(\overline{\mathbb F}_p)\). The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.

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Cite

@misc{OAI:Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026,
  author = {{OpenAI}},
  title = {{Integral Donovan Finiteness over Witt Vectors}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf}{OAI:Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.