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<!DOCTYPE html> <html lang="en"> <head> <title>isSCM -- checks whether a module or an ideal is sequentially Cohen-Macaulay</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="sequentially Cohen-Macaulay modules or ideals" href="index.html">SCMAlgebras</a> :: <a title="checks whether a module or an ideal is sequentially Cohen-Macaulay" href="_is__S__C__M.html">isSCM</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="_minimum__Dimension.html">next</a> | <a href="_filter__Ideal.html">previous</a> | <a href="_minimum__Dimension.html">forward</a> | <a href="_filter__Ideal.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>isSCM -- checks whether a module or an ideal is sequentially Cohen-Macaulay</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">isSCM M</code></dd> <dd><code class="language-macaulay2">isSCM I</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">M</span>, <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a finitely generated graded $S$-module, $S=K[x_1,\ldots,x_n]$, with $K$ a field or an ideal $I\subset S$</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">b</span>, <span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, whether the module $M$ or the graded algebra $S/I$ is sequentially Cohen-Macaulay</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Given a finitely generated graded $S$-module $M$, this method checks if the $i$th module of deficiency of $M$, $\omega^{i}(M)$, if non-zero, is Cohen-Macaulay of dimension $i$. For a homogeneous ideal $I\subset S$, the function checks if $\mathrm{depth} S/{I^{<i>}} \geq i+1$, where $I^{<i>}$ is the $i$th filter ideal.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S=QQ[x_1..x_5];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : M=coker matrix{{x_1*x_2,x_3*x_4,0,0},{0,x_1*x_5,x_2*x_4,0}};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : isSCM M o3 = false</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : S = QQ[x_1..x_10,y_1..y_10];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : E = {{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{6,7},{8,9},{8,10},{9,10}};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : J=ideal(for e in E list x_(e#0)*y_(e#1)-x_(e#1)*y_(e#0)); o6 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : isSCM J o7 = true</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">isSCM</span>:</h2> <ul> <li><kbd>isSCM(Ideal)</kbd></li> <li><kbd>isSCM(Module)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="checks whether a module or an ideal is sequentially Cohen-Macaulay" href="_is__S__C__M.html">isSCM</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">SCMAlgebras.m2:315:0</span>.</p> </div> </div> </div> </body> </html>