Result 199, Algebra

Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures

Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Modules describe how an algebra acts on a space; projective modules are especially tractable building blocks. Two unreviewed manuscripts report examples showing that even complete vanishing of certain extension groups need not force a module to be projective.

What changes?

The first manuscript constructs a finite-dimensional algebra over the rational-function field in three independent variables over the two-element field, and a finite-dimensional nonprojective module Z. All positive-degree extension groups, which describe higher extension relations, vanish both from Z to itself and from Z to the algebra. The second constructs a finite-dimensional symmetric algebra, a class with a strong duality property, over that same characteristic-two field, with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree.

What does that help mathematicians do?

The first module is also Gorenstein-projective, so the reported failure reaches that more structured class, contradicting both the Auslander-Reiten and Gorenstein-projective conjectures. The second example contradicts Tachikawa's second conjecture; its associated endomorphism algebra also contradicts the classical, generalized and strong Nakayama, Auslander-Gorenstein, and Wakamatsu tilting conjectures. Every counterexample persists under every extension of the base field. Thus, enlarging the field cannot repair these proposed implications for the constructed examples.

Are there practical applications?

The immediate value is foundational: these constructions challenge proposed ways to recognize projective modules from vanishing extension groups. Researchers seeking valid recognition criteria would need additional hypotheses that exclude the examples. The characteristic-two setting is important: the reported counterexamples defeat universal claims, but do not establish analogous failures over fields of other characteristics.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

An explicit counterexample to the Auslander-Reiten conjecture

September 23, 2026 30 pages Main result formalized in Lean

We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over $k=\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective left module Z such that $\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,Z)=\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,\Lambda)=0$ for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{An explicit counterexample to the Auslander--Reiten conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf}{OAI:An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026}},
  year = {2026}
}

A counterexample to Tachikawa's second conjecture

September 23, 2026 39 pages Main result formalized in Lean

We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over $k=\mathbb F_2(q,H_1,H_2)$ with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A counterexample to Tachikawa's second conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/199.md.

Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Auslander–Reiten conjecture predicts that a finitely generated module $M$ over an Artin algebra $A$ is projective if $\mathrm{Ext}^i_A(M,M\oplus A)=0$ for every $i>0$ . The formalized counterexample is a finite-dimensional algebra over $k=\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective Gorenstein-projective module with this vanishing.

It also establishes $A/\mathrm{rad}\,A\cong k^8$ , $(\mathrm{rad}\,A)^4\ne0$ , and preservation of the listed properties after every field extension. The Tachikawa companion is separate.

Tachikawa's second conjecture predicts that a finite-dimensional module over a finite-dimensional self-injective algebra is projective if all its positive-degree self-Ext groups vanish. The formalized counterexample gives a finite-dimensional symmetric algebra over $\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective module $M$ with $\mathrm{Ext}^i(M,M)=0$ for every $i>0$ . Symmetric algebras are self-injective, so this contradicts the conjecture.

The later field-extension, endomorphism-algebra, and related homological consequences are not included.

Comparator links

Result	Comparator statement
Auslander–Reiten counterexample	AuslanderReiten.lean
Tachikawa counterexample	Tachikawa.lean