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Math-2510 · mathematics · Algebra

Projectivity from simultaneous self-Ext and algebra-Ext vanishing

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every finite-dimensional algebra A over a field and every finite-dimensional left A-module M, does $Ext_A^i(M,M\oplus A)=0$ for all integers $i>0$ imply that M is projective?

Short English statement

For every finite-dimensional algebra A over a field and every finite-dimensional left A-module M, does $Ext_A^i(M,M\oplus A)=0$ for all integers $i>0$ imply that M is projective?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2510-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2510

The finite-dimensional nonprojective module Z has Ext^i(Z,Z)=Ext^i(Z,Lambda)=0 for every i>0 over the explicit finite-dimensional field algebra. Additivity gives the requested Ext^i(Z,Z plus Lambda)=0, a counterexample to the universal implication. lean/docs/199.md documents precisely this counterexample and its ancillary properties. Within these assumptions, this is a claimed negative answer to the stated question.

Projectivity from simultaneous self-Ext and algebra-Ext vanishing

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. Over $\mathbb F_2(q,H_1,H_2)$, an explicit finite algebra and nonprojective Gorenstein-projective module are claimed to have all positive self-Ext and Ext into the algebra equal to zero.

Mathematical statement of the claimed result

Result 1.
Over $k=\mathbb F_2(q,H_1,H_2)$, with algebraically independent indeterminates, the source claims a finite-dimensional algebra $\Lambda$ and nonprojective finite-dimensional left module $Z$ with \[ \operatorname{Ext}^i_\Lambda(Z,Z)=0= \operatorname{Ext}^i_\Lambda(Z,\Lambda)\qquad(i\ge1). \] The module is Gorenstein-projective, $\Lambda/\operatorname{rad}\Lambda\cong k^8$, and $(\operatorname{rad}\Lambda)^4\ne0$. All these conclusions persist after any extension of the coefficient field.

Outline of the argument

A parameterized complete resolution starts from a small algebra and is transformed by cone and triangular-module constructions. Powers of the transcendental parameters prevent unwanted recurrence, while uniform kernel calculations give total acyclicity and remove positive self-extensions; scalar extension preserves those calculations.

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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