# Normal scheme

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Concept in algebraic geometry

In [algebraic geometry](/source/Algebraic_geometry), an [algebraic variety](/source/Algebraic_varieties) or [scheme](/source/Scheme_(mathematics)) *X* is **normal** if it is normal at every point, meaning that the [local ring](/source/Local_ring_at_a_point) at the point is an [integrally closed domain](/source/Integrally_closed_domain).[1] An [affine variety](/source/Affine_variety) *X* (understood to be irreducible) is normal if and only if the ring *O*(*X*) of [regular functions](/source/Regular_function) on *X* is an integrally closed domain.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] A variety *X* over a field is normal if and only if every [finite](/source/Finite_morphism) [birational morphism](/source/Birational_geometry) from any variety *Y* to *X* is an [isomorphism](/source/Isomorphism).[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

Normal varieties were introduced by [Zariski](/source/Oscar_Zariski).[2]

## Geometric and algebraic interpretations of normality

A morphism of varieties is finite if the inverse image of every point is finite and the morphism is [proper](/source/Proper_morphism). A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve *X* in the affine plane *A*2 defined by *x*2 = *y*3 is not normal, because there is a finite birational morphism *A*1 → *X* (namely, *t* maps to (*t*3, *t*2)) which is not an isomorphism. By contrast, the affine line *A*1 is normal: it cannot be simplified any further by finite birational morphisms.

A normal complex variety *X* has the property, when viewed as a [stratified space](/source/Topologically_stratified_space) using the classical topology, that every link is connected. Equivalently, every complex point *x* has arbitrarily small neighborhoods *U* such that *U* minus the singular set of *X* is connected. For example, it follows that the nodal cubic curve *X* in the figure, defined by *y*2 = *x*2(*x* + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from *A*1 to *X* which is not an isomorphism; it sends two points of *A*1 to the same point in *X*.

Curve *y*2 = *x*2(*x* + 1)

More generally, a [scheme](/source/Scheme_(mathematics)) *X* is **normal** if each of its [local rings](/source/Local_ring)

- *O**X,x*

is an [integrally closed domain](/source/Integrally_closed_domain). That is, each of these rings is an [integral domain](/source/Integral_domain) *R*, and every ring *S* with *R* ⊆ *S* ⊆ Frac(*R*) such that *S* is finitely generated as an *R*-module is equal to *R*. (Here Frac(*R*) denotes the [field of fractions](/source/Field_of_fractions) of *R*.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to *X* is an isomorphism. For instance, in the case of the nodal cubic *X* in the figure, the local ring A = ( k [ x , y ] / ( y 2 − x 2 ( x + 1 ) ) ) ( x , y ) {\displaystyle A=\left(k[x,y]/(y^{2}-x^{2}(x+1))\right)_{(x,y)}} is not integrally closed in its field of fractions, since *y/x* is integral over *A* but is not in *A*. Therefore *X* is not normal at the point (0,0).[3]

An older notion is that a subvariety *X* of projective space is [linearly normal](/source/Linearly_normal#Projective_normality) if the linear system giving the embedding is complete. Equivalently, *X* ⊆ **P**n is not the linear projection of an embedding *X* ⊆ **P**n+1 (unless *X* is contained in a hyperplane **P**n). This is the meaning of "normal" in the phrases [rational normal curve](/source/Rational_normal_curve) and [rational normal scroll](/source/Rational_normal_scroll).

Every [regular scheme](/source/Glossary_of_scheme_theory#regular) is normal. Conversely, [Zariski](/source/Oscar_Zariski) showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes.[4][5] So, for example, every normal [curve](/source/Algebraic_curve) is regular.

## The normalization

Any [reduced scheme](/source/Reduced_scheme) *X* has a unique **normalization**: a normal scheme *Y* with an integral birational morphism *Y* → *X*. (For *X* a variety over a field, the morphism *Y* → *X* is finite, which is stronger than "integral".[6]) The normalization of a scheme of dimension 1 is regular, and the normalization of a scheme of dimension 2 has only isolated singularities. Normalization is not usually used for [resolution of singularities](/source/Resolution_of_singularities) for schemes of higher dimension.

To define the normalization, first suppose that *X* is an [irreducible](/source/Glossary_of_scheme_theory#irreducible) reduced scheme *X*. Every affine open subset of *X* has the form Spec *R* with *R* an [integral domain](/source/Integral_domain). Write *X* as a union of affine open subsets Spec *A*i. Let *B*i be the [integral closure](/source/Integral_closure) of *A*i in its fraction field. Then the normalization of *X* is defined by gluing together the affine schemes Spec *B*i.

If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.

### Examples

#### Normalization of a cusp

Consider the affine curve

C = Spec ( k [ x , y ] y 2 − x 5 ) {\displaystyle C={\text{Spec}}\left({\frac {k[x,y]}{y^{2}-x^{5}}}\right)}

with the cusp singularity at the origin. Its normalization can be given by the map

Spec ( k [ t ] ) → C {\displaystyle {\text{Spec}}(k[t])\to C}

induced from the algebra map

x ↦ t 2 , y ↦ t 5 {\displaystyle x\mapsto t^{2},y\mapsto t^{5}}

#### Normalization of axes in affine plane

For example,

X = Spec ( C [ x , y ] / ( x y ) ) {\displaystyle X={\text{Spec}}(\mathbb {C} [x,y]/(xy))}

is not an irreducible scheme since it has two components. Its normalization is given by the scheme morphism

Spec ( C [ x , y ] / ( x ) × C [ x , y ] / ( y ) ) → Spec ( C [ x , y ] / ( x y ) ) {\displaystyle {\text{Spec}}(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y))\to {\text{Spec}}(\mathbb {C} [x,y]/(xy))}

induced from the two quotient maps

C [ x , y ] / ( x y ) → C [ x , y ] / ( x , x y ) = C [ x , y ] / ( x ) {\displaystyle \mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(x,xy)=\mathbb {C} [x,y]/(x)}

C [ x , y ] / ( x y ) → C [ x , y ] / ( y , x y ) = C [ x , y ] / ( y ) {\displaystyle \mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(y,xy)=\mathbb {C} [x,y]/(y)}

#### Normalization of reducible projective variety

Similarly, for homogeneous irreducible polynomials f 1 , … , f k {\displaystyle f_{1},\ldots ,f_{k}} in a UFD, the normalization of

Proj ( k [ x 0 , … , x n ] ( f 1 ⋯ f k , g ) ) {\displaystyle {\text{Proj}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{(f_{1}\cdots f_{k},g)}}\right)}

is given by the morphism

Proj ( ∏ k [ x 0 … , x n ] ( f i , g ) ) → Proj ( k [ x 0 , … , x n ] ( f 1 ⋯ f k , g ) ) {\displaystyle {\text{Proj}}\left(\prod {\frac {k[x_{0}\ldots ,x_{n}]}{(f_{i},g)}}\right)\to {\text{Proj}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{(f_{1}\cdots f_{k},g)}}\right)}

## See also

- [Noether normalization lemma](/source/Noether_normalization_lemma)

- [Resolution of singularities](/source/Resolution_of_singularities)

## Notes

1. **[^](#cite_ref-1)** [Hartshorne, Robin](/source/Robin_Hartshorne) (1977), *[Algebraic Geometry](/source/Algebraic_Geometry_(book))*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 52, New York: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-387-90244-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90244-9), [MR](/source/MR_(identifier)) [0463157](https://mathscinet.ams.org/mathscinet-getitem?mr=0463157) p. 91

1. **[^](#cite_ref-2)** Zariski, Oscar (1939), "Some Results in the Arithmetic Theory of Algebraic Varieties.", *Amer. J. Math.*, **61** (2): 249–294, [doi](/source/Doi_(identifier)):[10.2307/2371499](https://doi.org/10.2307%2F2371499), [JSTOR](/source/JSTOR_(identifier)) [2371499](https://www.jstor.org/stable/2371499), [MR](/source/MR_(identifier)) [1507376](https://mathscinet.ams.org/mathscinet-getitem?mr=1507376) section III

1. **[^](#cite_ref-3)** Eisenbud, D. *Commutative Algebra* (1995). Springer, Berlin. Section 4.3

1. **[^](#cite_ref-4)** Zariski, Oscar (1939), "Some Results in the Arithmetic Theory of Algebraic Varieties.", *Amer. J. Math.*, **61** (2): 249–294, [doi](/source/Doi_(identifier)):[10.2307/2371499](https://doi.org/10.2307%2F2371499), [JSTOR](/source/JSTOR_(identifier)) [2371499](https://www.jstor.org/stable/2371499), [MR](/source/MR_(identifier)) [1507376](https://mathscinet.ams.org/mathscinet-getitem?mr=1507376) theorem 11

1. **[^](#cite_ref-5)** Eisenbud, D. *Commutative Algebra* (1995). Springer, Berlin. Theorem 11.5

1. **[^](#cite_ref-6)** Eisenbud, D. *Commutative Algebra* (1995). Springer, Berlin. Corollary 13.13

## References

- [Eisenbud, David](/source/David_Eisenbud) (1995), *Commutative algebra. With a view toward algebraic geometry.*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 150, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi](/source/Doi_(identifier)):[10.1007/978-1-4612-5350-1](https://doi.org/10.1007%2F978-1-4612-5350-1), [ISBN](/source/ISBN_(identifier)) [978-0-387-94268-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-94268-1), [MR](/source/MR_(identifier)) [1322960](https://mathscinet.ams.org/mathscinet-getitem?mr=1322960)

- [Hartshorne, Robin](/source/Robin_Hartshorne) (1977), *[Algebraic Geometry](/source/Algebraic_Geometry_(book))*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 52, New York: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-387-90244-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90244-9), [MR](/source/MR_(identifier)) [0463157](https://mathscinet.ams.org/mathscinet-getitem?mr=0463157), p. 91

- Zariski, Oscar (1939), "Some Results in the Arithmetic Theory of Algebraic Varieties.", *Amer. J. Math.*, **61** (2): 249–294, [doi](/source/Doi_(identifier)):[10.2307/2371499](https://doi.org/10.2307%2F2371499), [JSTOR](/source/JSTOR_(identifier)) [2371499](https://www.jstor.org/stable/2371499), [MR](/source/MR_(identifier)) [1507376](https://mathscinet.ams.org/mathscinet-getitem?mr=1507376)

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