Normality (commutative algebra)

In the mathematical branch of algebra , an integrity domain is called normal if it is completely closed in its quotient field . That means: is and all over , so is already . In general, any commutative ring is called normal if all of its local rings are normal domains. The two definitions are the same for health scopes. ${\displaystyle A}$ ${\displaystyle \alpha \in \mathrm {Quot} (A)}$ ${\displaystyle \alpha }$ ${\displaystyle A}$ ${\displaystyle \alpha \in A}$

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

properties

• Any factorial ring is normal.
• Every regular ring is normal.
• Locations of normal rings are back to normal.

If it is assumed that the ring is Noetherian , then:

• A normal ring is a finite product of normal integrity domains.
• A normal area of ​​integrity is the intersection of its localizations at prime ideals of height  1:

${\displaystyle A=\bigcap _{\operatorname {ht} {\mathfrak {p}}=1}A_{\mathfrak {p}}.}$

• The localizations at prime ideals of height 1 are discrete evaluation rings .

Examples

• The ring of whole numbers is normal. ${\displaystyle \mathbb {Z} }$
• The ring with the whole Gaussian number is also normal. ${\displaystyle \mathbb {Z} [i]=\{a+bi|a,b\in \mathbb {Z} \}}$ ${\displaystyle i^{2}=-1}$
• The ring for is not normal because i the quotient field of A is all about and A , but not in A is. ${\displaystyle A:=\mathbb {Z} [di]}$ ${\displaystyle d>1}$

Serre's normality criterion

A Noetherian ring is normal if and only if the conditions R ₁ and S _{2 are} met.

The regularity condition R _k for an integer means that the localizations at prime ideals of height are  regular . For a Noetherian integrity domain, R ₁ only means that the localizations at prime ideals of height 1 are discrete evaluation rings; for any Noetherian rings there is still reduction , i.e. H. the absence of nontrivial nilpotent elements , required. ${\displaystyle k\geq 0}$ ${\displaystyle \leq k}$

The Serre condition S _k for a natural number says that the depth of each local ring is greater than or equal to the minimum of its dimension and , in formulas ${\displaystyle k\geq 1}$ ${\displaystyle k}$

${\displaystyle \operatorname {tf} A_{\mathfrak {p}}\geq \min\{k,\dim A_{\mathfrak {p}}\}.}$

The combination of R ₁ and S ₂ can also be summarized as follows:

• For prime ideals of height the local ring is regular, i.e. H. a body or a discrete rating ring. ${\displaystyle \leq 1}$
• For prime ideals of height , the depth of the local ring is at least 2. ${\displaystyle \geq 2}$

In particular, the following applies: A one-dimensional Noetherian integrity domain is normal if and only if the localizations at the maximum ideals are discrete evaluation rings. Such rings are called Dedekind rings .

Applications

In algebraic geometry , a scheme is said to be normal when all of the local rings are normal. ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}_{X,x}}$

If there is any integral schema and the associated function field, then another schema , the normalization of , can be constructed as follows: If an open, affine subset, i.e. the spectrum of a ring , then form the entire closure of in . The spectra of the rings can be glued together to form a scheme . The morphism is induced by the inclusions . The normalization obtained in this way has the property of being regular in codimension 1. So if a curve is, it has no singularities. (Under mild conditions there is a resolution of the singularities in the sense of algebraic geometry.) ${\displaystyle X}$ ${\displaystyle K(X)}$ ${\displaystyle X^{norm}\rightarrow X}$ ${\displaystyle X}$ ${\displaystyle U\subset X}$ ${\displaystyle R}$ ${\displaystyle {\tilde {R}}}$ ${\displaystyle R}$ ${\displaystyle K(X)}$ ${\displaystyle {\tilde {R}}}$ ${\displaystyle X^{norm}}$ ${\displaystyle X^{norm}\rightarrow X}$ ${\displaystyle R\rightarrow {\tilde {R}}}$ ${\displaystyle X}$ ${\displaystyle X^{norm}}$ ${\displaystyle X^{norm}\rightarrow X}$

swell

• David Eisenbud, Commutative algebra with a view toward algebraic geometry . Springer-Verlag, New York 1995. ISBN 0-387-94269-6
• Hideyuki Matsumura, Commutative ring theory . Cambridge University Press, Cambridge 1989. ISBN 0-521-36764-6
• A. Grothendieck , J. Dieudonné : Éléments de géométrie algébrique . Publications mathématiques de l'IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967)