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class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Normalization of a cusp</span> </div> </a> <ul id="toc-Normalization_of_a_cusp-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normalization_of_axes_in_affine_plane" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Normalization_of_axes_in_affine_plane"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Normalization of axes in affine plane</span> </div> </a> <ul id="toc-Normalization_of_axes_in_affine_plane-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normalization_of_reducible_projective_variety" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Normalization_of_reducible_projective_variety"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Normalization of reducible projective variety</span> </div> </a> <ul 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<div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an <a href="/wiki/Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic variety</a> or <a href="/wiki/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="/wiki/Local_ring_at_a_point" class="mw-redirect" title="Local ring at a point">local ring</a> at the point is an <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">integrally closed domain</a>. An <a href="/wiki/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="/wiki/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="/wiki/Finite_morphism" title="Finite morphism">finite</a> <a href="/wiki/Birational_geometry" title="Birational geometry">birational morphism</a> from any variety <i>Y</i> to <i>X</i> is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>. </p><p>Normal varieties were introduced by <a href="/wiki/Oscar_Zariski" title="Oscar Zariski">Zariski</a> (<a href="#CITEREFZariski1939">1939</a>, section III). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Geometric_and_algebraic_interpretations_of_normality">Geometric and algebraic interpretations of normality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=1" title="Edit section: Geometric and algebraic interpretations of normality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A morphism of varieties is finite if the inverse image of every point is finite and the morphism is <a href="/wiki/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="/wiki/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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Newtonsche_Knoten.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Newtonsche_Knoten.png/220px-Newtonsche_Knoten.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Newtonsche_Knoten.png/330px-Newtonsche_Knoten.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Newtonsche_Knoten.png/440px-Newtonsche_Knoten.png 2x" data-file-width="598" data-file-height="585" /></a><figcaption>Curve <i>y</i><sup>2</sup> = <i>x</i><sup>2</sup>(<i>x</i> + 1)</figcaption></figure> <p>More generally, a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> <i>X</i> is <b>normal</b> if each of its <a href="/wiki/Local_ring" title="Local ring">local rings</a> </p> <dl><dd><i>O</i><sub><i>X,x</i></sub></dd></dl> <p>is an <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">integrally closed domain</a>. That is, each of these rings is an <a href="/wiki/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="/wiki/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="/wiki/Linearly_normal#Projective_normality" 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="/wiki/Rational_normal_curve" title="Rational normal curve">rational normal curve</a> and <a href="/wiki/Rational_normal_scroll" title="Rational normal scroll">rational normal scroll</a>. </p><p>Every <a href="/wiki/Glossary_of_scheme_theory#regular" 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="/wiki/Algebraic_curve" title="Algebraic curve">curve</a> is regular. </p> <div class="mw-heading mw-heading2"><h2 id="The_normalization">The normalization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=2" title="Edit section: The normalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any <a href="/wiki/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="/wiki/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="/wiki/Glossary_of_scheme_theory#irreducible" 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="/wiki/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="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Normalization_of_a_cusp">Normalization of a cusp</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=4" title="Edit section: Normalization of a cusp"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Consider the affine curve</p><blockquote><p><span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\text{Spec}}\left({\frac {k[x,y]}{y^{2}-x^{5}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/911a5814713125113003a841c21d5db37e3e058c" 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)}"></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"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}(k[t])\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eded942983b505e5335234195d66151f5d439206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.184ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}(k[t])\to C}"></span></p></blockquote><p>induced from the algebra map</p><blockquote><p><span class="mwe-math-element"><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\mapsto t^{2},y\mapsto t^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mi>y</mi> <mo stretchy="false">↦</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto t^{2},y\mapsto t^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1666ef22391dc1a8e8efec6371f67d3fd847278f" 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}}"></span></p></blockquote> <div class="mw-heading mw-heading4"><h4 id="Normalization_of_axes_in_affine_plane">Normalization of axes in affine plane</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=5" title="Edit section: Normalization of axes in affine plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> For example,</p><blockquote><p><span class="mwe-math-element"><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))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\text{Spec}}(\mathbb {C} [x,y]/(xy))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d237b10494d0c4f6e27c5cf076d49147557900" 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))}"></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"><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))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y))\to {\text{Spec}}(\mathbb {C} [x,y]/(xy))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/888da3a9b24086f646224774dbf14ba6c4c6e644" 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))}"></span></p></blockquote><p>induced from the two quotient maps</p><blockquote><p><span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(x,xy)=\mathbb {C} [x,y]/(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ffe6d0c0c59c546c502a64abee433741e46cbb" 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)}"></span></p></blockquote><blockquote><p> <span class="mwe-math-element"><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]/(y,xy)=\mathbb {C} [x,y]/(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(y,xy)=\mathbb {C} [x,y]/(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389c0fcd9a81de6f13b44e088a834573217d26ac" 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)}"></span></p></blockquote> <div class="mw-heading mw-heading4"><h4 id="Normalization_of_reducible_projective_variety">Normalization of reducible projective variety</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=6" title="Edit section: Normalization of reducible projective variety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Similarly, for homogeneous irreducible polynomials <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1},\ldots ,f_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8808d7cace871e6ef01dbd8e1a95e3ad0ae77ad4" 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}}"></span> in a UFD, the normalization of</p><blockquote><p><span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Proj</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Proj}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{(f_{1}\cdots f_{k},g)}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e722fff2f8e78abf0854bda871835d738620b52" 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)}"></span></p></blockquote><p>is given by the morphism</p><blockquote><p><span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Proj</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Proj</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <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> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e4c0469a45a6ebd2c9105760c52028cf8cfd29" 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)}"></span></p></blockquote> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Noether_normalization_lemma" title="Noether normalization lemma">Noether normalization lemma</a></li> <li><a href="/wiki/Resolution_of_singularities" title="Resolution of singularities">Resolution of singularities</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=8" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <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> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_scheme&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFEisenbud1995" class="citation cs2"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1995), <i>Commutative algebra. With a view toward algebraic geometry.</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 150, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/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="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94268-1" title="Special:BookSources/978-0-387-94268-1"><bdi>978-0-387-94268-1</bdi></a>, <a href="/wiki/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><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+algebra.+With+a+view+toward+algebraic+geometry.&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1995&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1322960%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-5350-1&rft.isbn=978-0-387-94268-1&rft.aulast=Eisenbud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+scheme" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <i><a href="/wiki/Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a></i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90244-9" title="Special:BookSources/978-0-387-90244-9"><bdi>978-0-387-90244-9</bdi></a>, <a href="/wiki/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><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=978-0-387-90244-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0463157%23id-name%3DMR&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+scheme" class="Z3988"></span>, p. 91</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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): 249–294, <a href="/wiki/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="/wiki/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="/wiki/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><span 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