TOPICS
Search

Tachikawa's Second Conjecture

Tachikawa's second conjecture (Tachikawa 1973) states that if Lambda is a self-injective Artin algebra and a finitely generated right Lambda-module X satisfies

Ext_Lambda^i(X,X)=0 for all i>0,

then X is projective. The Auslander-Reiten conjecture states that for any Artin algebra Lambda,

Ext_Lambda^i(X,X direct sum Lambda)=0 for all i>0

implies that X is projective.

Enomoto (2026) proved that Tachikawa's second conjecture for all self-injective Artin algebras over a fixed commutative Artinian ring implies the Auslander-Reiten conjecture for all Artin algebras over that ring. Together with known implications, this makes Tachikawa's second conjecture equivalent to the Auslander-Reiten, generalized Nakayama, Auslander-Gorenstein, Nakayama, and Gorenstein-projective conjectures in this setting.

Enomoto (2026) credits GPT-6 Astra with finding and drafting the proof, with Claude assisting in revision. As of Sep. 22, 2026, independent specialist review had not been reported. The equivalent conjectures remain open.

See also

Artinian Ring, Injective Module, Projective Module

Explore with Wolfram|Alpha

References

Auslander, M. and Reiten, I. "On a Generalized Version of the Nakayama Conjecture." Proc. Amer. Math. Soc. 52, 69-74, 1975. https://doi.org/10.2307/2040117.Enomoto, H. "Tachikawa's Second Conjecture Implies the Auslander-Reiten Conjecture." 14 Sep 2026. https://arxiv.org/abs/2609.19172.Tachikawa, H. Quasi-Frobenius Rings and Generalizations: QF-3 and QF-1 Rings (Notes by C. M. Ringel). Lecture Notes in Mathematics, Vol. 351. Berlin, Germany: Springer-Verlag, 1973.

Cite this as:

Weisstein, Eric W. "Tachikawa's Second Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TachikawasSecondConjecture.html

Subject classifications