On the homological conjectures for Artin algebras

Abstract

We prove that the Auslander--Reiten conjecture, the generalised Nakayama conjecture, the Nakayama conjecture, and each of the two Tachikawa conjectures are globally equivalent for Artin algebras over any fixed commutative artinian ring. For finite dimensional algebras over a fixed field, we also prove that the derived invariance of the Nakayama conjecture is equivalent to the truth of the Nakayama conjecture for all such algebras, answering a question raised by Chen and Xi.

10 pages. Version 2: Title changed; Rene Marczinzik joins as co-author. New results include the global implication from Tachikawa's first conjecture to the Auslander--Reiten conjecture, and the equivalence of the Nakayama conjecture with the Chen--Xi conjecture on derived invariance of finiteness of dominant dimension. Comments welcome!