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Is the Nakayama conjecture solved in the commutative case? It 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" I tried to look up but could not find if it is solved or not solved in the commutative case. Can someone provide a reference if it is solved in the commutative case. The wikipedia page does not say if it is solved in the commutative case. It is here: Thanks.

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up vote 1 down vote accepted

The following remark can be found in Morita Contexts, Idempotents, and Hochschild Cohomology — with Applications to Invariant Rings by Ragnar-Olaf Buchweitz (arXiv):

Remarks 3.2. (1) The conjectures (INC’), (INC) trivially hold if the algebras C,B involved are already commutative noetherian rings. However, there seems to be no real advantage gained in either (SNC) or (GNC) if one assumes that A is already commutative. In this sense, the aforementioned conjectures truly belong to the realm of (slightly) noncommutative algebra.

Here, SNC denotes the strong Nakayama conjecture and GNC the generalized Nakayama conjecture. For their meaning, see loc. cit.

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It says that INC(idempotent Nakayama Conjecture) is trivial, but it does not say that Nakayama conjecture is trivial or solved. – messi Jul 1 '12 at 9:07

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