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

Two papers claim Donovan's conjecture: blocks with bounded defect fall into finitely many types, for every prime.

2 papers September 24, 2026 Explained by AI

A finite group's representations in characteristic p split into pieces called blocks. Each block has a defect group, a small subgroup that measures how complicated it is. Donovan's conjecture says that, once the defect group is fixed, only finitely many kinds of block occur. Here two blocks are the same kind when their categories of modules match. These papers claim to prove this over every algebraically closed field, and in an integral form.

First step

1980

Jonathan Alperin's problem list, “Local representation theory” (Conjecture M)

Significance

Landmark

If the proofs hold, they settle a central finiteness conjecture of modular representation theory for every prime.

Trust

Claimed, not checked

Field

Algebra

2 papers

The explanation

Take the six symmetries of a triangle, the group S₃. Its group algebra is a six-dimensional space where group elements can be added and multiplied. Over a field where 5 = 0, this algebra splits into three blocks of dimensions 1, 1 and 4. The 4-dimensional block is all two-by-two matrices. Since 1 + 1 + 4 = 6, nothing is left over.

Over a field where 2 = 0, the same algebra splits differently, as 2 + 4. The 4 is again two-by-two matrices. The 2 is like numbers a + bx with x² = 0, which cannot be split further. That small piece carries the complication that the prime 2 brings, since 2 divides 6.

Each block has a defect group: a subgroup of prime-power size, determined up to conjugacy, that measures how complicated the block is. Matrix blocks have the trivial defect group. Two blocks count as the same kind, Morita equivalent, when their categories of modules match. Every matrix block is Morita equivalent to the plain field. So among blocks with trivial defect group, infinitely many groups give only one kind.

Donovan's conjecture asks whether this always happens. Fix a finite p-group. Can it be the defect group of only finitely many kinds of block, across all finite groupsSource 1? The conjecture is credited to Peter Donovan and appears as Conjecture M in Alperin's 1980 problem listSource 2. It was known for symmetric groups, by ScopesSource 3. Eaton and Livesey proved it for abelian 2-groupsSource 4. A recent preprint handled quaternion defect groupsSource 5.

The first paper claims the full conjecture over every algebraically closed field of characteristic pSource 6. That includes nonabelian defect groups and the prime 2. The second paper claims the stronger integral formSource 7. There the coefficients are p-adic-style numbers, Witt vectors, which remember more than the field doesSource 8. It also extends to any fixed complete coefficient ring with algebraically closed residue fieldSource 9.

Finiteness needs two kinds of controlSource 10. Blocks must have bounded size, and their multiplication tables must be definable over one finite field. If both hold, only finitely many tables can occurSource 11. The paper first bounds the sizes, through Cartan numbers, for blocks of quasisimple groupsSource 12 Source 13. Then it handles general groups by building up extensions.

The one obstacle is uniformity. In groups of Lie type, such as matrix groups over finite fields, some ingredients can grow without bound while the defect stays smallSource 14. The rank of the matrix group can grow too. The paper uses geometric tools to count the basic projective pieces of a block. A separate step reduces matrix groups of large rank to bounded cases.

How sure is this? No Lean formalization is listed for either paper. The claims rest on very long arguments that draw on deep geometric methods. No specialist review of these papers has been published.

What it doesn't show. This does not give an explicit list of the possible blocks or a usable bound on how many there are.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark if the proofs hold, but they are very long, have no Lean check, and no specialist review is published.

Words used

Group algebra: Formal sums of group elements with field coefficients, multiplied using the group's rule.
Block: One of the indecomposable pieces into which a group algebra splits.
Defect group: A prime-power subgroup attached to a block that measures how far it is from plain matrices.
Morita equivalence: Two algebras are Morita equivalent when their categories of modules match exactly.
Cartan numbers: Counts of how often each simple module appears inside each basic projective module of a block.
Witt vectors: A ring of p-adic-style numbers whose remainders modulo p form a given field.
Group of Lie type: A finite group built like a matrix group, such as invertible matrices over a finite field.

The original papers (2)

Donovan's Conjecture over Algebraically Closed Fields (PDF, opens in a new tab)
September 24, 2026
Integral Donovan Finiteness over Witt Vectors (PDF, opens in a new tab)
September 25, 2026

All quoted lines (14)

Donovan's conjecture asks whether a fixed finite $p$ -group can occur as the defect group of only finitely many blocks up to this equivalence.
Donovan's Conjecture over Algebraically Closed Fields Back to text
The conjecture is attributed to Peter Donovan and appears as Conjecture M in Alperin's account of problems in group representation theory.
Donovan's Conjecture over Algebraically Closed Fields Back to text
Scopes proved finiteness for symmetric-group blocks, and Kessar treated their double-cover families.
Donovan's Conjecture over Algebraically Closed Fields Back to text
Eaton–Livesey proved the field-valued conjecture for abelian $2$ -groups.
Donovan's Conjecture over Algebraically Closed Fields Back to text
The August 2026 preprint proves integral finiteness for quaternion defect groups and deduces finiteness for all tame blocks over $k$ .
Donovan's Conjecture over Algebraically Closed Fields Back to text
In particular, over this field $K$ , Donovan's conjecture holds for every finite $p$ -group, including nonabelian groups.
Donovan's Conjecture over Algebraically Closed Fields Back to text
As $G$ ranges over all finite groups and $b$ ranges over all blocks of $\cO G$ whose defect groups have order at most $M$ , the algebras $\cO G b$ belong to only finitely many $\cO$ -linear Morita equivalence classes.
Integral Donovan Finiteness over Witt Vectors Back to text
Over $\cO$ it concerns integral module categories, including their lattice structure, and hence is stronger than the corresponding assertion over the residue field.
Integral Donovan Finiteness over Witt Vectors Back to text
The finiteness conclusion also passes to a fixed complete coefficient ring with algebraically closed residue field.
Integral Donovan Finiteness over Witt Vectors Back to text
The general theorem requires both multiplicity control and control of field of definition.
Donovan's Conjecture over Algebraically Closed Fields Back to text
If both the dimension and the size of a finite field of definition are bounded, only finitely many multiplication tables can occur.
Donovan's Conjecture over Algebraically Closed Fields Back to text
The proof first establishes a uniform Cartan bound for quasisimple blocks of bounded defect.
Donovan's Conjecture over Algebraically Closed Fields Back to text
The Cartan estimate bounds the sizes of the basic algebras; the comparison controls their multiplication and descent data.
Donovan's Conjecture over Algebraically Closed Fields Back to text
In a group of Lie type, finite tori can grow even when the chosen block has small defect; a bound for a principal block with small Sylow subgroup does not address this case.
Donovan's Conjecture over Algebraically Closed Fields Back to text

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