Geodesic
Graded rings ofcohomological dimension 2
Glasgow mathematical journal, Tome 42 (2000) no. 3, pp. 455-468

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Let A be a noetherian connectedgraded ring with a balanced dualizing complex R. If A has cohomological dimension and Krull dimension2, 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 aself-injective artinian quotient ring;(5) A has a residuecomplex.(1,3,4) generalize a result of Levasseur [7, 5.13] and (5) generalizes aresult of Ajitabh [1, 3.12]. 
Wu, Q.S.; Zhang, J. J. Graded rings ofcohomological dimension 2. Glasgow mathematical journal, Tome 42 (2000) no. 3, pp. 455-468. doi: 10.1017/S0017089500030123
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