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<!DOCTYPE html> <html lang="en"> <head> <title>isCM -- whether a ring or module is 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="Finds the depth of a module or ideal, and systems of parameters in an ideal" href="index.html">Depth</a> :: <a title="whether a ring or module is Cohen-Macaulay" href="_is__C__M.html">isCM</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="_is__Regular__Sequence.html">next</a> | <a href="_inhomogeneous__System__Of__Parameters.html">previous</a> | <a href="_is__Regular__Sequence.html">forward</a> | <a href="_inhomogeneous__System__Of__Parameters.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>isCM -- whether a ring or module is Cohen-Macaulay</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">isCM(A)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">A</span>, a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">Ring</a> or <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">Module</a></span></li> </ul> </li> <li>Outputs: <ul> <li><span><a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean</a></span></li> </ul> </li> </ul> <div> <h2>Description</h2> This command merely checks if the depth of <span class="tt">A</span> equals the Krull dimension of <span class="tt">A</span>. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : A = ZZ/2[x,y,z];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : isCM(A) o2 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : A = ZZ/2[x,y]/(x^2,x*y);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : isCM(A) o4 = false</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : A = ZZ/101[a_1,a_2,b_1,b_2,c_1]/ideal(a_1*b_1,a_2*b_2,b_1*c_1);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : isCM(A) o6 = false</code></pre> </td> </tr> </table> <p>This symbol is provided by the package <a title="Finds the depth of a module or ideal, and systems of parameters in an ideal" href="index.html">Depth</a>.</p> </div> <div> <h2>Caveat</h2> Typically when one thinks of a Cohen-Macaulay ring or module, one is in the local case. Since the local case is not yet implemented into Macaulay 2, we compute over the ideal generated by by <a title="the list of generators of a ring" href="../../Macaulay2Doc/html/_generators_lp__Ring_rp.html">generators(Ring)</a>. </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">isCM</span>:</h2> <ul> <li><kbd>isCM(Module)</kbd></li> <li><kbd>isCM(Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="whether a ring or module is Cohen-Macaulay" href="_is__C__M.html">isCM</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">Depth.m2:835:0</span>.</p> </div> </div> </div> </body> </html>