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Math-2516 · mathematics · Algebra

Integral block finiteness over a fixed complete valuation ring

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix a prime p, a complete characteristic-zero discrete valuation ring O with algebraically closed residue field of characteristic p, and $M\geq 1$. Are there only finitely many O-linear Morita classes of finite-group block orders OGb whose residue defect groups have order at most M?

Short English statement

Fix a prime p, a complete characteristic-zero discrete valuation ring O with algebraically closed residue field of characteristic p, and $M\geq 1$. Are there only finitely many O-linear Morita classes of finite-group block orders OGb whose residue defect groups have order at most M?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2516-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2516

The fixed-complete-coefficient-ring corollary states exactly O-linear bounded-residue-defect Morita finiteness for every fixed complete characteristic-zero DVR with algebraically closed characteristic-p residue field. Within these assumptions, this is a claimed affirmative answer to the stated question.

Integral block finiteness over a fixed complete valuation ring

OpenAI's claim

OpenAI presents a proof in the selected manuscript. 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.

Mathematical statement of the claimed result

Result 1.
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 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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