Algebras that satisfy auslander's condition on vanishing of cohomology

TY - JOUR

T1 - Algebras that satisfy auslander's condition on vanishing of cohomology

AU - Christensen, Lars Winther

AU - Holm, Henrik Granau

PY - 2010/5

Y1 - 2010/5

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.

U2 - 10.1007/s00209-009-0500-4

DO - 10.1007/s00209-009-0500-4

M3 - Journal article

SN - 0025-5874

VL - 265

SP - 21

EP - 40

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1

ER -