Result 203, Algebra

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

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.

Proof

The bigger picture

Why it matters

A finite group's representations can be organized into blocks, algebraic pieces describing how the group acts on vector spaces. The manuscripts report that bounding a block's defect leaves only finitely many essentially different module theories.

What changes?

Defect groups are associated subgroups whose orders are powers of p. For every prime p, fixed algebraically closed field of characteristic p, and bound on defect-group order, the claim is finitely many Morita classes: blocks grouped by equivalent module theories. The integral claim gives the same finiteness over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings. Neither claim requires abelian defect groups, and both include p = 2.

What does that help mathematicians do?

This would rule out an infinite family of genuinely different block module theories with uniformly bounded defect-group order over a fixed coefficient field or ring. Researchers could therefore organize module-theoretic questions around finitely many representatives, even while the underlying finite groups vary without restriction. The conclusion concerns equivalence of module theories, not a finite list of groups, and the stated finiteness supplies no numerical count.

Are there practical applications?

The immediate value is foundational: it would constrain the possible structures in the representation theory of finite groups. The integral version extends that constraint to coefficient rings connecting characteristic zero with characteristic p through reduction, including the specified Witt-vector ring. This is a structural classification result, rather than a demonstrated computational method or practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Donovan's Conjecture over Algebraically Closed Fields

September 24, 2026 94 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 25, 2026 22 pages

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(F‾p)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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.