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Integral Donovan Finiteness over Witt Vectors

OpenAI · September 25, 2026

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Abstract

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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Possibly relevant literature (found by the agent, not checked)

Florian Eisele and John W. MacQuarrie, Finiteness properties of profinite blocks and blocks with defect group $\mathbb Z_p^n$ (arXiv:2609.32587, 2026-09-26). Shows that blocks with defect group $\mathbb Z_p^n$ are Morita equivalent to twisted algebras $O_\alpha[[\mathbb Z_p^n \rtimes E]]$ with $E$ a $p'$-group and deduces Donovan's conjecture for such profinite blocks, close in method to the paper's integral treatment of twisted $p'$-kernel algebras and crossed orders over $O$; it is not in the paper's bibliography.
Charles W. Eaton and Michael Livesey, Donovan's conjecture and extensions by the centralizer of a defect group (arXiv:2006.11173, 2020-06-19). Gives a normal-subgroup reduction for Donovan's conjecture over a discrete valuation ring with integral finiteness for defect groups $Q_8 \times C_{2^n}$ and $Q_8 \times Q_8$, bearing on the paper's normal-subgroup reductions to crossed orders and its removal of the normal $p'$-kernel; it is not in the paper's bibliography.

Elsewhere

Emergent Mind: family 203

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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}
}