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Are there any connected graded Noetherian algebra of infinite injective dimension but has a balanced dualizing complex?

Thanks a lot.

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  • $\begingroup$ All commutative connected graded noetherian algebras of finite Krull dimension have balanced dualizing complexes, by van den Bergh's criterion [they always have property $\chi$ and Grothendieck's theorem implies their local dimension is equal to their Krull dimension]. Pick any one that doesn't have finite injective dimension. $\endgroup$
    – Pablo Zadunaisky
    Commented Oct 14, 2013 at 7:04
  • $\begingroup$ Any commutative connected graded noetherian algebra is a quotient of polynomial algebra in finite indeterminants, and so it has a balanced dualizing complex. However do you have any explicite such examples? Thank you very much! $\endgroup$
    – jason_zhou
    Commented Oct 14, 2013 at 8:28
  • $\begingroup$ $k[x,y]/(x^2,xy,y^2)$. See also en.wikipedia.org/wiki/Gorenstein_ring. $\endgroup$
    – user91132
    Commented Oct 14, 2013 at 8:41

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