One Hat Cyber Team

One Hat Cyber Team

Your IP : 216.73.216.135

Server IP : 194.44.31.54

Server : Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64

Server Software : Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k

PHP Version : 5.6.40

Buat File | Buat Folder

Dir : ~/usr/share/doc/Macaulay2/Isomorphism/html/

Edit File: _isomorphism.html

<!DOCTYPE html>
<html lang="en">
  <head>
    <title>isomorphism -- retrieve an isomorphism of modules</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="probabilistic test for isomorphism of modules" href="index.html">Isomorphism</a> :: <a title="retrieve an isomorphism of modules" href="_isomorphism.html">isomorphism</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="_is__Isomorphic.html">previous</a> | forward | <a href="_is__Isomorphic.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a>      </div>
    </div>
    <hr>
    <div>
      <h1>isomorphism -- retrieve an isomorphism of modules</h1>
      <ul>
        <li>
          <dl class="element">
            <dt>Usage: </dt>
            <dd><code class="language-macaulay2">f = isomorphism(N, M)</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>, </span></li>
            <li><span><span class="tt">N</span>, <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, </span></li>
          </ul>
        </li>
        <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:          <ul>
            <li><span><span class="tt">Homogeneous</span><span class="tt"> => </span><span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value true</span>, if false, allow an inhomogeneous isomorphism; </span></li>
            <li><span><span class="tt">Verbose</span><span class="tt"> => </span><span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, whether to print debugging information; </span></li>
            <li><span><span class="tt">Strict</span><span class="tt"> => </span><span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, whether to force strict equality of degrees; </span></li>
            <li><span><span class="tt">Tries</span><span class="tt"> => </span><span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, <span>default value null</span>, the number of attempts at generating random map; </span></li>
            <li><span><span class="tt">Strategy</span> (missing documentation)
<span class="tt"> => </span><span class="tt">...</span>, <span>default value null</span>, <span></span></span></li>
          </ul>
        </li>
        <li>Outputs:          <ul>
            <li><span><span class="tt">f</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, an isomorphism $M \to N$; </span></li>
          </ul>
        </li>
      </ul>
      <div>
        <h2>Description</h2>
        <div>
          <p>This method retrieves the isomorphism between the given modules. Note that this isomorphism is already cached if a previous call to <a title="probabilistic test for isomorphism of modules" href="_is__Isomorphic.html">isIsomorphic</a> returned true, but otherwise that function is called.</p>
        </div>
        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i1 : S = ZZ/32003[x_0..x_3]

o1 = S

o1 : PolynomialRing</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i2 : m = random(S^3, S^{4:-2});

3      4
o2 : Matrix S  &lt;-- S</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i3 : A = random(target m, target m);

3      3
o3 : Matrix S  &lt;-- S</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i4 : B = random(source m, source m);

4      4
o4 : Matrix S  &lt;-- S</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i5 : N = coker(A*m*B);</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i6 : M = coker m;</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i7 : f = isomorphism(N, M)

o7 = | 1     4078 4892  |
     | -2922 6959 -89   |
     | -5122 -889 -1064 |

o7 : Matrix N &lt;-- M</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i8 : isIsomorphism f

o8 = true</code></pre>
            </td>
          </tr>
        </table>
      </div>
      <div>
        <h2>See also</h2>
        <ul>
          <li><span><a title="probabilistic test for isomorphism of modules" href="_is__Isomorphic.html">isIsomorphic</a> -- probabilistic test for isomorphism of modules</span></li>
          <li><span><a title="whether a map is an isomorphism" href="../../Macaulay2Doc/html/_is__Isomorphism.html">isIsomorphism</a> -- whether a map is an isomorphism</span></li>
        </ul>
      </div>
      <div>
        <div class="waystouse">
          <h2>Ways to use <span class="tt">isomorphism</span>:</h2>
          <ul>
            <li><kbd>isomorphism(Module,Module)</kbd></li>
          </ul>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="retrieve an isomorphism of modules" href="_isomorphism.html">isomorphism</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">Isomorphism.m2:528:0</span>.</p>
        </div>
      </div>
    </div>
  </body>

</html>