Nakayama's conjecture

In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by Nakayama (1958). The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander and Reiten (1975). Leuschke & Huneke (2004) proved some cases of the generalized Nakayama conjecture.

Nakayama's conjecture states that if all the modules of a minimal injective resolution of an Artin algebra R are injective and projective, then R is self-injective.

References

• Auslander, Maurice; Reiten, Idun (1975), "On a generalized version of the Nakayama conjecture", Proceedings of the American Mathematical Society, 52 (1): 69–74, doi:10.2307/2040102, ISSN 0002-9939, JSTOR 2040102, MR 0389977
• Leuschke, Graham J.; Huneke, Craig (2004), "On a conjecture of Auslander and Reiten", Journal of Algebra, 275 (2): 781–790, arXiv:math/0305001, doi:10.1016/j.jalgebra.2003.07.018, ISSN 0021-8693, MR 2052636, S2CID 14400955
• Nakayama, Tadasi (1958), "On algebras with complete homology", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 22: 300–307, doi:10.1007/BF02941960, ISSN 0025-5858, MR 0104718, S2CID 116606420