Let A be a noetherian connected
graded ring with a balanced dualizing complex R. If A has cohomological dimension and Krull dimension
2, then
(1) R is Auslander;
(2) \rm{Cdim} M=\rm{Kdim}
M for all noetherian graded A-modules M.
In particular, ifA
is AS-Gorenstein of injective and Krull dimension 2, then
(3)A
is Auslander-Gorenstein;
(4) A is 2-pure with a
self-injective artinian quotient ring;
(5) A has a residue
complex.
(1,3,4) generalize a result of Levasseur [7, 5.13] and (5) generalizes a
result of Ajitabh [1, 3.12].