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<title>Normal scheme</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Normal scheme</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an <a href="Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic variety</a> or <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> <i>X</i> is <b>normal</b> if it is normal at every point, meaning that the <a href="Local_ring_at_a_point" class="mw-redirect" title="Local ring at a point">local ring</a> at the point is an <a href="Integrally_closed_domain" title="Integrally closed domain">integrally closed domain</a>. An <a href="Affine_variety" title="Affine variety">affine variety</a> <i>X</i> (understood to be irreducible) is normal if and only if the ring <i>O</i>(<i>X</i>) of <a href="Regular_function" class="mw-redirect" title="Regular function">regular functions</a> on <i>X</i> is an integrally closed domain. A variety <i>X</i> over a field is normal if and only if every <a href="Finite_morphism" title="Finite morphism">finite</a> <a href="Birational_geometry" title="Birational geometry">birational morphism</a> from any variety <i>Y</i> to <i>X</i> is an <a href="Isomorphism" title="Isomorphism">isomorphism</a>.

</p><p>Normal varieties were introduced by <a href="Oscar_Zariski" title="Oscar Zariski">Zariski</a> (<a href="#CITEREFZariski1939">1939</a>, section III).

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<div class="mw-heading mw-heading2"><h2 id="Geometric_and_algebraic_interpretations_of_normality">Geometric and algebraic interpretations of normality</h2></div>

<p>A morphism of varieties is finite if the inverse image of every point is finite and the morphism is <a href="Proper_morphism" title="Proper morphism">proper</a>. A morphism of varieties

is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve <i>X</i> in the affine plane <i>A</i><sup>2</sup> defined by <i>x</i><sup>2</sup> = <i>y</i><sup>3</sup> is not normal, because there is a finite birational morphism <i>A</i><sup>1</sup> → <i>X</i>

(namely, <i>t</i> maps to (<i>t</i><sup>3</sup>, <i>t</i><sup>2</sup>)) which is not an isomorphism. By contrast, the affine line <i>A</i><sup>1</sup> is normal: it cannot be simplified any further by finite birational morphisms.

</p><p>A normal complex variety <i>X</i> has the property, when viewed as a <a href="Topologically_stratified_space" class="mw-redirect" title="Topologically stratified space">stratified space</a> using the classical topology, that every link is connected. Equivalently, every complex point <i>x</i> has arbitrarily small neighborhoods <i>U</i> such that <i>U</i> minus

the singular set of <i>X</i> is connected. For example, it follows that the nodal cubic curve <i>X</i> in the figure, defined by <i>y</i><sup>2</sup> = <i>x</i><sup>2</sup>(<i>x</i> + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from <i>A</i><sup>1</sup> to <i>X</i> which is not an isomorphism; it sends two points of <i>A</i><sup>1</sup> to the same point in <i>X</i>.

</p>

<p>More generally, a <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> <i>X</i> is <b>normal</b> if each of its <a href="Local_ring" title="Local ring">local rings</a>

</p>

<p>is an <a href="Integrally_closed_domain" title="Integrally closed domain">integrally closed domain</a>. That is, each of these rings is an <a href="Integral_domain" title="Integral domain">integral domain</a> <i>R</i>, and every ring <i>S</i> with <i>R</i> ⊆ <i>S</i> ⊆ Frac(<i>R</i>) such that <i>S</i> is finitely generated as an <i>R</i>-module is equal to <i>R</i>. (Here Frac(<i>R</i>) denotes the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of <i>R</i>.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to <i>X</i> is an isomorphism.

</p><p>An older notion is that a subvariety <i>X</i> of projective space is <a href="Linearly_normal" class="mw-redirect" title="Linearly normal">linearly normal</a> if the linear system giving the embedding is complete. Equivalently, <i>X</i> ⊆ <b>P</b><sup>n</sup> is not the linear projection of an embedding <i>X</i> ⊆ <b>P</b><sup>n+1</sup> (unless <i>X</i> is contained

in a hyperplane <b>P</b><sup>n</sup>). This is the meaning of "normal" in the phrases <a href="Rational_normal_curve" title="Rational normal curve">rational normal curve</a> and <a href="Rational_normal_scroll" title="Rational normal scroll">rational normal scroll</a>.

</p><p>Every <a href="Glossary_of_scheme_theory" class="mw-redirect" title="Glossary of scheme theory">regular scheme</a> is normal. Conversely, <a href="#CITEREFZariski1939">Zariski (1939</a>, theorem 11) showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> So, for example, every normal <a href="Algebraic_curve" title="Algebraic curve">curve</a> is regular.

</p>

<div class="mw-heading mw-heading2"><h2 id="The_normalization">The normalization</h2></div>

<p>Any <a href="Reduced_scheme" class="mw-redirect" title="Reduced scheme">reduced scheme</a> <i>X</i> has a unique <b>normalization</b>: a normal scheme <i>Y</i> with an integral birational morphism <i>Y</i> → <i>X</i>. (For <i>X</i> a variety over a field, the morphism <i>Y</i> → <i>X</i> is finite, which is stronger than "integral".<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>) 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 <a href="Resolution_of_singularities" title="Resolution of singularities">resolution of singularities</a> for schemes of higher dimension.

</p><p>To define the normalization, first suppose that <i>X</i> is an <a href="Glossary_of_scheme_theory" class="mw-redirect" title="Glossary of scheme theory">irreducible</a> reduced scheme <i>X</i>. Every affine open subset of <i>X</i> has the form Spec <i>R</i> with <i>R</i> an <a href="Integral_domain" title="Integral domain">integral domain</a>. Write <i>X</i> as a union of affine open subsets Spec <i>A</i><sub>i</sub>. Let <i>B</i><sub>i</sub> be the <a href="Integral_closure" class="mw-redirect" title="Integral closure">integral closure</a> of <i>A</i><sub>i</sub> in its fraction field. Then the normalization of <i>X</i> is defined by gluing together the affine schemes

Spec <i>B</i><sub>i</sub>.

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

</p>

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Normalization_of_a_cusp">Normalization of a cusp</h4></div><p>

Consider the affine curve</p><blockquote><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\text{Spec}}\left({\frac {k[x,y]}{y^{2}-x^{5}}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle C={\text{Spec}}\left({\frac {k[x,y]}{y^{2}-x^{5}}}\right)}</annotation>

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</math></span><img src="./911a5814713125113003a841c21d5db37e3e058c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.598ex; height:6.343ex;" alt="{\displaystyle C={\text{Spec}}\left({\frac {k[x,y]}{y^{2}-x^{5}}}\right)}" loading="lazy"></span></p></blockquote><p>with the cusp singularity at the origin. Its normalization can be given by the map</p><blockquote><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}(k[t])\to C}">

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</math></span><img src="./1666ef22391dc1a8e8efec6371f67d3fd847278f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.535ex; height:3.009ex;" alt="{\displaystyle x\mapsto t^{2},y\mapsto t^{5}}" loading="lazy"></span></p></blockquote>

<div class="mw-heading mw-heading4"><h4 id="Normalization_of_axes_in_affine_plane">Normalization of axes in affine plane</h4></div><p>

For example,</p><blockquote><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\text{Spec}}(\mathbb {C} [x,y]/(xy))}">

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<annotation encoding="application/x-tex">{\displaystyle X={\text{Spec}}(\mathbb {C} [x,y]/(xy))}</annotation>

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</math></span><img src="./14d237b10494d0c4f6e27c5cf076d49147557900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.485ex; height:2.843ex;" alt="{\displaystyle X={\text{Spec}}(\mathbb {C} [x,y]/(xy))}" loading="lazy"></span></p></blockquote><p>is not an irreducible scheme since it has two components. Its normalization is given by the scheme morphism</p><blockquote><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y))\to {\text{Spec}}(\mathbb {C} [x,y]/(xy))}">

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</math></span><img src="./888da3a9b24086f646224774dbf14ba6c4c6e644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.731ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y))\to {\text{Spec}}(\mathbb {C} [x,y]/(xy))}" loading="lazy"></span></p></blockquote><p>induced from the two quotient maps</p><blockquote><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(x,xy)=\mathbb {C} [x,y]/(x)}">

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</math></span><img src="./03ffe6d0c0c59c546c502a64abee433741e46cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.764ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(x,xy)=\mathbb {C} [x,y]/(x)}" loading="lazy"></span></p></blockquote><blockquote><p>

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</math></span><img src="./389c0fcd9a81de6f13b44e088a834573217d26ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.416ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(y,xy)=\mathbb {C} [x,y]/(y)}" loading="lazy"></span></p></blockquote>

<div class="mw-heading mw-heading4"><h4 id="Normalization_of_reducible_projective_variety">Normalization of reducible projective variety</h4></div><p>

Similarly, for homogeneous irreducible polynomials <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\ldots ,f_{k}}">

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</math></span><img src="./8808d7cace871e6ef01dbd8e1a95e3ad0ae77ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.6ex; height:2.509ex;" alt="{\displaystyle f_{1},\ldots ,f_{k}}" loading="lazy"></span> in a UFD, the normalization of</p><blockquote><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Proj}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{(f_{1}\cdots f_{k},g)}}\right)}">

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</math></span><img src="./7e722fff2f8e78abf0854bda871835d738620b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.628ex; height:6.509ex;" alt="{\displaystyle {\text{Proj}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{(f_{1}\cdots f_{k},g)}}\right)}" loading="lazy"></span></p></blockquote><p>is given by the morphism</p><blockquote><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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)}">

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<annotation encoding="application/x-tex">{\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)}</annotation>

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</math></span><img src="./65e4c0469a45a6ebd2c9105760c52028cf8cfd29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.581ex; height:6.509ex;" alt="{\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)}" loading="lazy"></span></p></blockquote>

<ul><li><a href="Noether_normalization_lemma" title="Noether normalization lemma">Noether normalization lemma</a></li>

<li><a href="Resolution_of_singularities" title="Resolution of singularities">Resolution of singularities</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Eisenbud, D. <i>Commutative Algebra</i> (1995). Springer, Berlin. Theorem 11.5</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Eisenbud, D. <i>Commutative Algebra</i> (1995). Springer, Berlin. Corollary 13.13</span>

</li>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFEisenbud1995" class="citation cs2"><a href="David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1995), <i>Commutative algebra. With a view toward algebraic geometry.</i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 150, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-5350-1">10.1007/978-1-4612-5350-1</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94268-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1322960">1322960</a></cite></li>

<li><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <i><a href="Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a></i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90244-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></cite>, p. 91</li>

<li><cite id="CITEREFZariski1939" class="citation cs2">Zariski, Oscar (1939), "Some Results in the Arithmetic Theory of Algebraic Varieties.", <i>Amer. J. Math.</i>, <b>61</b> (2): <span class="nowrap">249–</span>294, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371499">10.2307/2371499</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371499">2371499</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1507376">1507376</a></cite></li></ul>

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