Algebras that satisfy Auslander's condition on vanishing of cohomology

Details Algebras that satisfy Auslander's condition on vanishing of cohomology Algebras that satisfy Auslander's condition on vanishing of cohomology

2008-01-02

arxiv.org/abs/0801.0401v2

Mathematics

Rings and Algebras

Final version, to appear in Math. Z. 20 pp

Scientific paper

Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture - by a 2003 counterexample due to Jorgensen and Sega - motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.

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