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Math-2422 · mathematics · Algebra

Donovan’s Finiteness Conjecture for Fixed Defect Groups

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Fix a prime $p$, an algebraically closed field $k$ of characteristic $p$, and a finite $p$-group $P$. Among all finite groups $G$, are there only finitely many Morita-equivalence classes of blocks of $kG$ with defect group isomorphic to $P$?

Formulation references: [S1], [S2].

Short English statement

Fix a prime $p$, an algebraically closed field $k$ of characteristic $p$, and a finite $p$-group $P$. Among all finite groups $G$, are there only finitely many Morita-equivalence classes of blocks of $kG$ with defect group isomorphic to $P$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2422-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2422

Fixing a defect group fixes its order. Finiteness for all blocks of bounded defect order over every algebraically closed characteristic-$p$ field therefore gives the requested fixed-defect-group Morita finiteness. The Witt-vector theorem proves an additional integral version.

Donovan’s Finiteness Conjecture for Fixed Defect Groups

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Blocks with bounded defect-group size are claimed to fall into finitely many Morita classes over any fixed algebraically closed field of positive characteristic; Bounded defect is claimed to yield finitely many integral Morita classes over Witt vectors, and over any fixed complete valuation coefficient ring of mixed characteristic, provided its residue field is algebraically closed. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.
Fix prime $p$, an arbitrary algebraically closed field $K$ of characteristic $p$, and an integer $M\ge1$. The source claims \[ \#\{K[G]b:\ G\text{ finite},\ b\text{ block idempotent},\ |D_b|\le M\}/\text{Morita equivalence}<\infty. \] Here $D_b$ is a defect group and equivalence is $K$-linear. Taking $M=|P|$ includes every block with a prescribed finite $p$-group $P$ as defect group, whether $P$ is abelian or not. The theorem itself allows all such $K$, despite the narrower paper title.
Result 2.
For $\mathcal O=W(\overline{\mathbb F}_p)$ and integer $M\ge1$, the source claims finitely many $\mathcal O$-linear Morita classes among $\mathcal O[G]b$, as $G$ ranges over finite groups and blocks $b$ have defect-group order at most $M$. Its complete-ring corollary states more generally: fix a complete discretely valued ring $\mathcal R$, whose fraction field has characteristic zero and whose residue field $\ell$ has characteristic $p$ and is algebraically closed. Then \[ \#\{\mathcal R[G]b:\ G\text{ finite},\ b\text{ primitive central idempotent},\ |D_{\bar b}|\le M\}/\mathcal R\text{-linear Morita equivalence}<\infty. \] Only the block and finite group vary; the coefficient ring remains fixed.

Outline of the argument

The argument separately controls the size of a basic block algebra and its finite field of definition. Uniform Cartan and Frobenius bounds are combined with a finiteness criterion, then scalar descent transfers the result from algebraic closures of prime fields to the chosen coefficient field.

The proof combines integral bounds for basic orders with a bound on their Morita--Frobenius behavior. Rationality and Cartan controls feed the integral finiteness criterion, and lifting from the residue field handles fixed ramified coefficient rings as well as Witt vectors.

Formal verification

Family 203 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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