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<!DOCTYPE html> <html lang="en"> <head> <title>isSyzygy -- Tests if a module is a d-th syzygy</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="produces an ideal with three generators whose 2nd syzygy module is isomorphic to a given module" href="index.html">Bruns</a> :: <a title="Tests if a module is a d-th syzygy" href="_is__Syzygy.html">isSyzygy</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> next | <a href="_evans__Griffith.html">previous</a> | forward | <a href="_evans__Griffith.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>isSyzygy -- Tests if a module is a d-th syzygy</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">b = isSyzygy(M,d)</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>, over a polynomial ring.</span></li> <li><span><span class="tt">d</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, positive</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>, returns <span class="tt">true</span> if M is a d-th syzygy, and <span class="tt">false</span> otherwise.</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This algorithm is based upon the methods described in the book of Evans and Griffith (<em>Syzygies</em>. London Mathematical Society Lecture Note Series, 106. Cambridge University Press, Cambridge, 1985.)</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : kk=ZZ/32003 o1 = kk o1 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S=kk[a..d] o2 = S o2 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : F=res (ideal vars S)^2 1 10 20 15 4 o3 = S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 o3 : ChainComplex</code></pre> </td> </tr> </table> <div> <p><b>NOTE:</b> We are viewing a syzygy module as a cokernel of an appropriate map.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : isSyzygy(coker F.dd_3,3) -- the cokernel defined by the 3rd map is not a 3rd syzygy o4 = false</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : isSyzygy(coker F.dd_4,3) -- the cokernel defined by the 4th map is a 3rd syzygy o5 = true</code></pre> </td> </tr> </table> <div> <p>This function is called within <a title="Reduces the rank of a syzygy" href="_evans__Griffith.html">evansGriffith</a>.</p> <p></p> </div> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">isSyzygy</span>:</h2> <ul> <li><kbd>isSyzygy(Module,ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Tests if a module is a d-th syzygy" href="_is__Syzygy.html">isSyzygy</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">Bruns.m2:463:0</span>.</p> </div> </div> </div> </body> </html>