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