1 · The essentials
These manuscripts give explicit examples that contradict several proposed rules in algebra. The rules connect the way mathematical objects interact with whether those objects have a particular simpler structure. In the main example, every interaction test specified by the rule gives zero, yet the expected structure is absent. A companion manuscript gives another example under a different set of requirements and links it to several further failures. Both constructions use a particular number system in which adding one to one gives zero. The counterexamples remain valid whenever that number system is enlarged by a field extension. Both manuscripts are identified as Lean-formalized, although the supplied abstracts do not specify the exact coverage of the formal verification.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a checklist intended to guarantee that a piece of furniture can be taken apart into standard components. These examples are like furniture that passes every check on the prescribed list but still cannot be taken apart in the required way. The mathematical checks cover every positive degree, not merely a finite sample.
Why it matters
The examples show that the stated vanishing conditions do not force projectivity, even in the particular settings described by the manuscripts. The companion construction also connects this failure to several other conjectures through an associated endomorphism algebra. Persistence under every extension of the base field makes field enlargement insufficient to remove these counterexamples.
2 · Going further
The main manuscript addresses the Auslander–Reiten conjecture for Artin algebras. It constructs a finite-dimensional algebra over the rational-function field and a finite-dimensional left module satisfying for every , while is not projective. Thus the specified absence of higher extensions does not force projectivity.
The module is also Gorenstein-projective. According to the main manuscript, this makes the same construction a counterexample to the Gorenstein-projective conjecture as well. The argument described in the abstract is constructive: exhibit an algebra and a module that satisfy the vanishing requirements but fail the predicted conclusion. The supplied text does not give the defining relations or the computations establishing those properties.
The companion manuscript constructs a finite-dimensional symmetric algebra over the same field, together with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. This supplies its counterexample to Tachikawa’s second conjecture. An associated endomorphism algebra also supplies counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture.
Both manuscripts state that their counterexamples persist after every extension of . Both are labeled Lean-formalized in the supplied source; the abstracts themselves do not distinguish which individual construction steps or companion consequences are covered by that formalization.
Useful notions
- Fields and field extensions
- Finite-dimensional algebras and modules
- Projective modules
- Extension groups and positive degrees
What was known before / what is new
Before
The source identifies the Auslander–Reiten, Gorenstein-projective, and Tachikawa’s second conjectures as the principal targets. It also names five further conjectures addressed through an associated endomorphism algebra. It gives no account of earlier partial results or the history of these conjectures.
New
The main manuscript supplies an explicit nonprojective module with both stated families of extension groups vanishing in all positive degrees; that module is also Gorenstein-projective. The companion supplies a symmetric-algebra example and an associated endomorphism algebra addressing the additional conjectures. All the stated counterexamples persist under every extension of the chosen base field.
Limits and assumptions
The constructions begin over the specific characteristic-two field , with finite-dimensional algebras and finite-dimensional modules. Persistence under every extension of this field is not a statement about arbitrary base fields or other characteristics. Both manuscripts are identified as Lean-formalized, but the supplied source does not delineate the formalization’s coverage of every listed consequence. It supplies neither detailed proofs nor statements of what remains open under other hypotheses.
Mini glossary
- Module —
- A module is an algebraic object on which an algebra acts.
- Projective module —
- A projective module is one for which maps into a quotient module can be lifted to the module before taking the quotient.
- Extension groups —
- The groups record higher extension information between modules, indexed by degree .
- Field extension —
- A field extension is a larger field containing the original field with its operations preserved.
- Endomorphism algebra —
- An endomorphism algebra consists of the structure-preserving maps from a module to itself, with composition as multiplication.
Keywords : Counterexamples · Extension groups · Projectivity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
An explicit counterexample to the Auslander-Reiten conjecture
23/09/2026We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over and a finite-dimensional nonprojective left module Z such that 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.
A counterexample to Tachikawa's second conjecture
23/09/2026We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over 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.