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Math-2525 · mathematics · Algebra

Self-Ext vanishing over symmetric algebras

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every finite-dimensional symmetric algebra A over a field, must each finite-dimensional module M with $Ext_A^i(M,M)=0$ for every integer $i>0$ be projective?

Short English statement

For every finite-dimensional symmetric algebra A over a field, must each finite-dimensional module M with $Ext_A^i(M,M)=0$ for every integer $i>0$ be projective?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2525-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2525

The main theorem gives a finite-dimensional symmetric algebra over F_2(q,H_1,H_2) and a finite-dimensional nonprojective module with all positive self-Ext zero. lean/docs/199.md directly documents this Tachikawa main counterexample; later homological consequences are unnecessary. Within these assumptions, this is a claimed negative answer to the stated question.

Self-Ext vanishing over symmetric algebras

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. A finite-dimensional symmetric algebra is claimed to carry a nonprojective finite module without positive self-extensions, refuting Tachikawa second-conjecture rigidity.

Mathematical statement of the claimed result

Result 1.
Take $k=\mathbb F_2(q,H_1,H_2)$, with three algebraically independent indeterminates. The source claims a finite-dimensional symmetric unital algebra $A$ and a nonprojective finite-dimensional left module $M$ satisfying \[ \operatorname{Ext}^i_A(M,M)=0\qquad(i>0). \] For every extension field $K/k$, put $A_K=K\otimes_kA$, $M_K=K\otimes_kM$, and $\Gamma_K=\operatorname{End}_{A_K}(A_K\oplus M_K)^{\mathrm{op}}$. The homological corollary claims that its minimal injective resolution has only projective terms but infinite injective dimension; also a nonzero simple $S$ satisfies \[ \operatorname{Ext}^i_{\Gamma_K}(S,\Gamma_K)=0\qquad(i\ge0). \] It supplies a projective-injective Wakamatsu tilting module that is not tilting. The finitistic corollary provides a left module $C_n$, finite-dimensional as a vector space, for which $\operatorname{pd}_{\Gamma_K}C_n=n$ for each $n\ge1$, hence both left finitistic dimensions are infinite.

Outline of the argument

The construction strengthens complete-resolution vanishing on both sides of degree zero before passing to a trivial extension. A transfer calculation then produces a symmetric algebra and preserves the nonprojective module together with its positive Ext vanishing.

Formal verification

For family 199, the documented entries cover the characteristic-two rational-function-field Auslander--Reiten example, including field-extension persistence, and a separate symmetric Tachikawa counterexample. The listed Comparator files are AuslanderReiten.lean, Tachikawa.lean. Named declarations include OAI.ArExplicit.Statement.main, OAI.Tachikawa.main_theorem. 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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