Algebras that satisfy Auslander's condition on vanishing of cohomology

Lars Winther Christensen; Henrik Holm

doi:10.1007/s00209-009-0500-4

Algebras that satisfy Auslander's condition on vanishing of cohomology

Lars Winther Christensen, Henrik Holm

Mathematics and Statistics

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36 Scopus citations

Abstract

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 Şega-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.

Original language	English
Pages (from-to)	21-40
Number of pages	20
Journal	Mathematische Zeitschrift
Volume	265
Issue number	1
DOIs	https://doi.org/10.1007/s00209-009-0500-4
State	Published - May 2010

Keywords

AB ring
AC ring
Conjectures of Auslander, Reiten, and Tachikawa
G-dimension
Gorenstein algebra

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10.1007/s00209-009-0500-4

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abstract = "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 {\c S}ega-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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N2 - 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 Şega-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.

AB - 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 Şega-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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KW - AC ring

KW - Conjectures of Auslander, Reiten, and Tachikawa

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KW - Gorenstein algebra

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Algebras that satisfy Auslander's condition on vanishing of cohomology

Abstract

Keywords

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