Donovan's conjecture

Donovan's conjecture predicts that, for a fixed finite p-group D, the blocks of group algebras of finite groups with defect group D fall into only finitely many Morita equivalence classes. Two manuscripts dated 24 and 25 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim proofs over every algebraically closed field of characteristic p and, in an integral form, over the Witt vectors of the algebraic closure of F_p and over every complete discrete valuation ring of characteristic zero with algebraically closed residue field. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

Background

As recounted in the introductions of the two manuscripts:
- The conjecture is attributed to Peter Donovan and appears as Conjecture M in Alperin's account of problems in group representation theory. Its Cartan-boundedness part grows out of a question of Brauer, and Hiss emphasized the additional rationality obstruction: bounded defect and Cartan invariants give Morita finiteness when the blocks are defined over a common finite field. Kessar separated these issues through Morita–Frobenius numbers.
- Scopes proved finiteness for blocks of symmetric groups and Kessar for their double covers. Jost's reductions bound the rank for unipotent blocks of general linear groups, and Hiss and Kessar developed reductions for unipotent blocks of classical groups.
- Eaton and Livesey proved the conjecture over the field for abelian 2-groups. Eaton, Eisele and Livesey proved an integral finiteness criterion, bounded defect, Cartan sum and integral Morita–Frobenius number, which gives integral finiteness for abelian 2-groups. Farrell and Kessar bounded Morita–Frobenius numbers of quasisimple blocks, An and Eaton treated extraspecial groups of order p³, and Eaton treated Suzuki 2-groups.
- For quaternion defect groups, Eaton, Eisele, Kessar, Linckelmann and Schaeffer Fry proved integral finiteness in an August 2026 preprint, arXiv 2608.12052, and deduced finiteness for all tame blocks over the field. Külshammer introduced crossed-product methods into the reduction, and Eisele developed integral crossed-product reductions and proved that integral block Picard groups are finite.

The original results

Claim (over fields). Theorem 1.1 of Donovan's Conjecture over Algebraically Closed Fields states that for every prime p, every algebraically closed field K of characteristic p and every M, blocks of finite groups over K with defect groups of order at most M fall into finitely many K-linear Morita equivalence classes, nonabelian defect groups and p = 2 included.
Claim (integral). Theorem 1.1 of Integral Donovan Finiteness over Witt Vectors states the same over W(F̄_p), and its Corollary 1.2 extends it to every fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, ramified rings included.
Method, as described: the field proof first bounds Cartan invariants of quasisimple blocks of bounded defect uniformly. For groups of Lie type with fixed root data this uses projective characters built from tilting objects following Eteve, with Lang local systems removing dependence on the finite torus. For classical groups of unbounded rank it uses Harish-Chandra moves on cones of projective characters and a direct treatment of cuspidal endpoints. A general block is then reduced to crossed extensions of finitely many integral base orders, and Frobenius comparisons compatible with Sylow actions and inner factor systems, together with finite integral Picard groups, leave finitely many crossed systems.
Sources. The manuscripts Donovan's Conjecture over Algebraically Closed Fields and Integral Donovan Finiteness over Witt Vectors. The collection contains no Lean formalization or scope note for these results.
Catalogue status. The open-problems catalogue 10m-u/open-problems-catalog lists the problem, entry Q2013 Finitely many block types, among its excluded entries as "claimed in openai/math", with the note "upstream claim, not independently verified", checked 18 October 2026 at commit 637649b.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 18 October 2026)
2026-09-26	F. Eisele and J. W. MacQuarrie, arXiv 2609.32587v1, Finiteness properties of profinite blocks and blocks with defect group Z_pⁿ	Related work	Every block of a profinite group with topologically finitely generated defect group is isomorphic to a block of a virtually pro-p group, and every block with defect group Z_pⁿ is Morita equivalent to a twisted completed group algebra of Z_pⁿ ⋊ E. The paper asks for which pro-p groups D there are only finitely many blocks with defect group D up to Morita equivalence, noting that this contains Donovan's conjecture as a special case. It does not cite the OpenAI manuscripts.	arXiv preprint, version 1.
2026-10-16	J. Jardine, Lattice24, referee reviews rank_127 (field) and rank_445 (integral) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Both reviews report no gap. For the field manuscript, the reviewer states that the finiteness of the integral base orders depends on the cited theorems of Eaton, Eisele and Livesey and of Eisele, whose exact hypotheses it could not check, and that it spot-checked but did not fully verify two estimates behind the Cartan bound. For the integral manuscript, it notes that one transport step is asserted for conjugate blocks without re-running the comparison explicitly.	Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.

Notes

The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.