Algebras that satisfy Auslander's condition on vanishing of cohomology

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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 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 languageEnglish
Pages (from-to)21-40
Number of pages20
JournalMathematische Zeitschrift
Volume265
Issue number1
DOIs
StatePublished - May 2010

Keywords

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

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