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About: Serre's criterion for normality     Goto   Sponge   NotDistinct   Permalink

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In algebra, Serre's criterion for normality, introduced by Jean-Pierre Serre, gives necessary and sufficient conditions for a commutative Noetherian ring A to be a normal ring. The criterion involves the following two conditions for A: * is a regular local ring for any prime ideal of height ≤ k. * for any prime ideal . The statement is: * A is a reduced ring hold. * A is a normal ring hold. * A is a Cohen–Macaulay ring hold for all k. Items 1, 3 trivially follow from the definitions. Item 2 is much deeper.