Normal scheme

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A scheme all local rings (cf. Local ring) of which are normal (that is, reduced and integrally closed in their ring of fractions). A normal scheme is locally irreducible; for such a scheme the concepts of a connected component and an irreducible component are the same. The set of singular points of a Noetherian normal scheme has codimension greater than 1. The following normality criterion holds [1]: A Noetherian scheme is normal if and only if two conditions are satisfied: 1) for any point of codimension the local ring is regular (cf. Regular ring (in commutative algebra)); and 2) for any point of codimension the depth of the ring (cf. Depth of a module) is greater than 1. Every reduced scheme has a normal scheme canonically connected with it (normalization). The -scheme is integral, but not always finite over . However, if is excellent (see Excellent ring), for example, if is a scheme of finite type over a field, then is finite over .

References

 [1] J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1975)