Normal scheme

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 $ X $ is normal if and only if two conditions are satisfied: 1) for any point $ x \in X $ of codimension $ \leq 1 $ the local ring $ {\mathcal O} _ {X,x} $ is regular (cf. Regular ring (in commutative algebra)); and 2) for any point $ x \in X $ of codimension $ > 1 $ the depth of the ring (cf. Depth of a module) $ {\mathcal O} _ {X,x} $ is greater than 1. Every reduced scheme $ X $ has a normal scheme $ X ^ \nu $ canonically connected with it (normalization). The $ X $- scheme $ X ^ \nu $ is integral, but not always finite over $ X $. However, if $ X $ is excellent (see Excellent ring), for example, if $ X $ is a scheme of finite type over a field, then $ X ^ \nu $ is finite over $ X $.

References

[1]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1975) MR0201468 Zbl 0296.13018

Comments

A normalization of an irreducible algebraic variety $ X $ is an irreducible normal variety $ X ^ \nu $ together with a regular mapping $ \nu : X ^ \nu \rightarrow X $ that is finite and a birational isomorphism.

For an affine irreducible algebraic variety, $ X ^ \nu $ is the integral closure of the ring $ A ( X) $ of regular functions on $ X $ in its field of fractions. The normalization has the following universality properties. Let $ X $ be an integral scheme (i.e. $ X $ is both reduced and irreducible, or, equivalently, $ {\mathcal O} _ {X} ( U) $ is an integral domain for all open $ U $ in $ X $). For every normal integral scheme $ Z $ and every dominant morphism $ f : Z \rightarrow X $( i.e. $ f ( Z) $ is dense in $ X $), $ f $ factors uniquely through the normalization $ X ^ \nu \rightarrow X $. So also Normal analytic space.

Let $ X $ be a curve and $ x $ a, possibly singular, point on $ X $. Let $ X ^ \nu \rightarrow X $ be the normalization of $ X $ and $ \overline{x}\; _ {1} \dots \overline{x}\; _ {n} $ the inverse images of $ x $ in $ X ^ \nu $. These points are called the branches of $ X $ passing through $ x $. The terminology derives from the fact that the $ \overline{x}\; _ {i} $ can be identified (in the case of varieties over $ \mathbf R $ or $ \mathbf C $) with the "branches" of $ X $ passing through $ x $. More precisely, if the $ U _ {i} $ are sufficiently small complex or real neighbourhoods of the $ x _ {i} $, then some neighbourhood of $ x $ is the union of the branches $ \nu ( U _ {i} ) $. Let $ T _ {i} $ be the tangent space at $ \overline{x}\; _ {i} $ to $ X ^ \nu $. Then $ ( d \nu ) ( \overline{x}\; _ {i} ) ( T _ {i} ) $ is some linear subspace of the tangent space to $ X $ at $ x $. It will be either a line or a point. In the first case the branch $ \overline{x}\; _ {i} $ is called linear. The point $ ( 0 , 0 ) $ on $ y ^ {2} = x ^ {3} + x ^ {2} $ is an example of a point with two linear branches (with tangents $ y = x $, $ y = - x $), and the point $ ( 0 , 0 ) $ on $ y ^ {2} = x ^ {3} $ gives an example of a two-fold non-linear branch.

$$ \begin{array}{lc} X ^ \nu &{} \\ {} &\downarrow {\nu } \\ X &{} \\ \end{array} \ \ \ \ \ \begin{array}{l} X ^ \nu \\ \downarrow {\nu } \\ X \\ \end{array} $$

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001
[a2]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Sect. II.5 (Translated from Russian) MR0366917 Zbl 0284.14001
[a3]	H. Matsumura, "Commutative algebra" , Benjamin (1970) MR0266911 Zbl 0211.06501