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<!DOCTYPE html> <html lang="en"> <head> <title>seminormalize -- seminormalize a reduced ring</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="a package used to seminormalize rings" href="index.html">Seminormalization</a> :: <a title="seminormalize a reduced ring" href="_seminormalize.html">seminormalize</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_seminormalize_lp..._cm__Variable_eq_gt..._rp.html">next</a> | <a href="_ring__To__Algebra__Map.html">previous</a> | <a href="_seminormalize_lp..._cm__Variable_eq_gt..._rp.html">forward</a> | <a href="_ring__To__Algebra__Map.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>seminormalize -- seminormalize a reduced ring</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">seminormalize(S)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">S</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, </span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="controls what strategy is used in calls to integralClosure" href="_better__Normalization__Map_lp..._cm__Strategy_eq_gt..._rp.html">Strategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span>controls what strategy is used in calls to integralClosure</span></span></li> <li><span><a title="set the name for new variables created by the function" href="_seminormalize_lp..._cm__Variable_eq_gt..._rp.html">Variable</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value Yy</span>, <span>set the name for new variables created by the function</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, the first entry of which is the seminormalization of the ring, the second and thirds are maps between the ring, its seminormalization and its normalization</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This seminormalizes a reduced ring and outputs a list, the first entry of which is the seminormalized ring, the second is the map from the ring to its seminormalization, and finally the map from the seminormalization to its normalization. In our first example, the cusp, the seminormalization and normalization are isomorphic.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x,y]/ideal(x^3 - y^2);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : L = seminormalize(R) QQ[Yy ..Yy ] 0 2 o2 = {---------------------------------------, map 2 2 (Yy - Yy , Yy Yy - Yy , Yy - Yy Yy ) 2 1 1 2 0 1 0 2 ------------------------------------------------------------------------ QQ[Yy ..Yy ] 0 2 (---------------------------------------, R, {Yy , Yy }), map 2 2 1 0 (Yy - Yy , Yy Yy - Yy , Yy - Yy Yy ) 2 1 1 2 0 1 0 2 ------------------------------------------------------------------------ QQ[Yy , x..y] 0,0 (------------------------------------, 2 2 (Yy y - x , Yy x - y, Yy - x) 0,0 0,0 0,0 ------------------------------------------------------------------------ QQ[Yy ..Yy ] 0 2 ---------------------------------------, {y, x, Yy })} 2 2 0,0 (Yy - Yy , Yy Yy - Yy , Yy - Yy Yy ) 2 1 1 2 0 1 0 2 o2 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : L#0 QQ[Yy ..Yy ] 0 2 o3 = --------------------------------------- 2 2 (Yy - Yy , Yy Yy - Yy , Yy - Yy Yy ) 2 1 1 2 0 1 0 2 o3 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : target(L#2) QQ[Yy , x..y] 0,0 o4 = ------------------------------------ 2 2 (Yy y - x , Yy x - y, Yy - x) 0,0 0,0 0,0 o4 : QuotientRing</code></pre> </td> </tr> </table> <div> <p>The previous example seminormalized a non-seminormal ring. Let's try a seminormal ring (the pinch point).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : R = QQ[x,y,z]/ideal(x^2*y-z^2);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : L = seminormalize(R) QQ[Yy ..Yy ] QQ[Yy ..Yy ] 0 2 0 2 o6 = {------------, map (------------, R, {Yy , Yy , Yy }), map 2 2 2 2 1 2 0 Yy Yy - Yy Yy Yy - Yy 1 2 0 1 2 0 ------------------------------------------------------------------------ QQ[Yy , x..z] QQ[Yy ..Yy ] 0,0 0 2 (-------------------------------------, ------------, {z, x, y})} 2 2 2 (Yy z - x*y, Yy x - z, Yy - y) Yy Yy - Yy 0,0 0,0 0,0 1 2 0 o6 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : L#0 QQ[Yy ..Yy ] 0 2 o7 = ------------ 2 2 Yy Yy - Yy 1 2 0 o7 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : target(L#2) QQ[Yy , x..z] 0,0 o8 = ------------------------------------- 2 (Yy z - x*y, Yy x - z, Yy - y) 0,0 0,0 0,0 o8 : QuotientRing</code></pre> </td> </tr> </table> <div> <p>We conclude with an example of a ring where the seminormalization, the normalization and the ring itself are all are distinct, the tacnode.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i9 : R = QQ[x,y]/ideal(y*(y-x^2));</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : L = seminormalize(R) QQ[Yy ..Yy ] QQ[Yy ..Yy ] 0 1 0 1 2 o10 = {------------, map (------------, R, {Yy + Yy , Yy }), map Yy Yy Yy Yy 0 1 0 0 1 0 1 ----------------------------------------------------------------------- QQ[Yy ..Yy ] QQ[Yy0, Yy1, Yy2] 0 1 (------------------------------------, ------------, {Yy1, Yy0})} 2 Yy Yy (Yy2 - Yy2, Yy1*Yy2 - Yy1, Yy0*Yy2) 0 1 o10 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : L#0 QQ[Yy ..Yy ] 0 1 o11 = ------------ Yy Yy 0 1 o11 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : target(L#2) QQ[Yy0, Yy1, Yy2] o12 = ------------------------------------ 2 (Yy2 - Yy2, Yy1*Yy2 - Yy1, Yy0*Yy2) o12 : QuotientRing</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">seminormalize</span>:</h2> <ul> <li><kbd>seminormalize(Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="seminormalize a reduced ring" href="_seminormalize.html">seminormalize</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Seminormalization.m2:732:0</span>.</p> </div> </div> </div> </body> </html>