One Hat Cyber Team
Your IP :
216.73.216.216
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
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
Kronecker
/
html
/
View File Name :
_decompose__Module.html
<!DOCTYPE html> <html lang="en"> <head> <title>decomposeModule -- decompose a module into a direct sum of simple 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="Kronecker and rational normal forms" href="index.html">Kronecker</a> :: <a title="decompose a module into a direct sum of simple modules" href="_decompose__Module.html">decomposeModule</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="_double__Dual__Map.html">next</a> | <a href="_decompose__Graded__Module.html">previous</a> | <a href="_double__Dual__Map.html">forward</a> | <a href="_decompose__Graded__Module.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>decomposeModule -- decompose a module into a direct sum of simple modules</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">(N,f) = decomposeModule 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>, a module over a ring k[x,y]/(x^2,y^2)</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">N</span>, <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a module, isomorphic to <span class="tt">M</span> which is a direct sum of simple modules</span></li> <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 from <span class="tt">N</span> to <span class="tt">M</span></span></li> </ul> </li> </ul> <div> <h2>Description</h2> This function decomposes a module into a direct sum of simple modules, given some fairly strong assumptions on the ring which acts on the ring which acts on the module. This ring must only have two variables, and the square of each of those variables must kill the module. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : Q = ZZ/101[x,y] o1 = Q o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : R = Q/(x^2,y^2) o2 = R o2 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : M = coker random(R^5, R^8 ** R^{-1}) o3 = cokernel | 24x-36y -29x-24y -18x-13y 45x-34y 39x+43y 40x+11y 2x+29y 27x-22y | | -30x-29y -38x-16y -43x-15y -48x-47y -17x-11y 46x-28y -47x+15y 32x-9y | | 19x+19y 39x+21y -28x-47y 47x+19y 48x+36y x-3y -37x-13y -32x-20y | | -10x-29y 34x+19y 38x+2y -16x+7y 35x+11y 22x-47y -10x+30y 24x-30y | | -8x-22y -47x-39y 16x+22y 15x-23y -38x+33y -23x-7y -18x+39y -48x-15y | 5 o3 : R-module, quotient of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : (N,f) = decomposeModule M o4 = (cokernel | y x 0 0 0 0 0 0 |, | -25 -49 -25 15 1 |) | 0 0 x 0 y 0 0 0 | | 28 -45 29 30 -39 | | 0 0 0 y x 0 0 0 | | -16 43 -43 -47 -24 | | 0 0 0 0 0 x 0 y | | -14 -37 25 -19 26 | | 0 0 0 0 0 0 y x | | 1 0 0 0 0 | o4 : Sequence</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : components N o5 = {cokernel | y x |, cokernel | x 0 y |, cokernel | x 0 y |} | 0 y x | | 0 y x | o5 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : ker f == 0 o6 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : coker f == 0 o7 = true</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">decomposeModule</span>:</h2> <ul> <li><kbd>decomposeModule(Module)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="decompose a module into a direct sum of simple modules" href="_decompose__Module.html">decomposeModule</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">Kronecker.m2:1187:0</span>.</p> </div> </div> </div> </body> </html>