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Many local rings have geometric interpretations. Cohen–Macaulay rings correspond to equi-dimensionality, and regular local rings correspond to non-singularity, but what is a geometric interpretation of Gorenstein local rings? I am not able to find one, so suggestions are most welcome!

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Cohen-Macaulay is much stronger than the equidimensionality which is just a topological property. –  user18119 Dec 21 '12 at 13:52

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