<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>subquotient -- make a subquotient module</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Subring__Limit.html">next</a> | <a href="_submatrix_lp__Matrix_cm__Visible__List_cm__Visible__List_rp.html">previous</a> | <a href="___Subring__Limit.html">forward</a> | <a href="_submatrix_lp__Matrix_cm__Visible__List_cm__Visible__List_rp.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>subquotient -- make a subquotient module</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>subquotient(g,r)</tt><br/><tt>subquotient(M,g,r)</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>g</tt>, the matrix of generators</span></li>
<li><span><tt>r</tt>, the matrix of relations, with the same target as <tt>g</tt></span></li>
<li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, if given, must be the common target of g and r</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, the image of <tt>g</tt> in the cokernel of <tt>r</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>If <tt>g</tt> is omitted, then the identity map on the target of <tt>r</tt> is used as <tt>g</tt>, and the cokernel of <tt>r</tt> is returned. If <tt>r</tt> is omitted (but not the comma), then a zero map to the target of <tt>g</tt> is used as <tt>r</tt>, and the image of <tt>g</tt> is returned. If both are omitted, then <tt>M</tt> is returned.</p>
<p>See <a href="_subquotient_spmodules.html" title="the way Macaulay2 represents modules">subquotient modules</a> for an overview of subquotient modules in Macaulay2.</p>
<p>The general form in which modules are represented in Macaulay2 is as subquotients, and subquotient modules are often returned as values of computations, as in the example below.</p>
<table class="examples"><tr><td><pre>i1 : R = ZZ/101[a..d]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : M = kernel vars R ++ cokernel vars R
o2 = subquotient ({1} | -b 0 -c 0 0 -d 0 |, {1} | 0 0 0 0 |)
{1} | a -c 0 0 -d 0 0 | {1} | 0 0 0 0 |
{1} | 0 b a -d 0 0 0 | {1} | 0 0 0 0 |
{1} | 0 0 0 c b a 0 | {1} | 0 0 0 0 |
{0} | 0 0 0 0 0 0 1 | {0} | a b c d |
5
o2 : R-module, subquotient of R</pre>
</td></tr>
<tr><td><pre>i3 : generators M
o3 = {1} | -b 0 -c 0 0 -d 0 |
{1} | a -c 0 0 -d 0 0 |
{1} | 0 b a -d 0 0 0 |
{1} | 0 0 0 c b a 0 |
{0} | 0 0 0 0 0 0 1 |
5 7
o3 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i4 : relations M
o4 = {1} | 0 0 0 0 |
{1} | 0 0 0 0 |
{1} | 0 0 0 0 |
{1} | 0 0 0 0 |
{0} | a b c d |
5 4
o4 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i5 : M === subquotient(generators M, relations M)
o5 = true</pre>
</td></tr>
<tr><td><pre>i6 : prune M,
o6 = (cokernel {2} | 0 0 0 0 c 0 0 d |, )
{2} | 0 0 0 0 a d 0 0 |
{2} | 0 0 0 0 -b 0 d 0 |
{2} | 0 0 0 0 0 b a 0 |
{2} | 0 0 0 0 0 -c 0 a |
{2} | 0 0 0 0 0 0 -c -b |
{0} | d c b a 0 0 0 0 |
o6 : Sequence</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_subquotient_spmodules.html" title="the way Macaulay2 represents modules">subquotient modules</a> -- the way Macaulay2 represents modules</span></li>
<li><span><a href="_is__Subquotient_lp__Module_cm__Module_rp.html" title="check whether a module is a subquotient of another">isSubquotient</a> -- check whether a module is a subquotient of another</span></li>
<li><span><a href="_ambient.html" title="ambient free module of a subquotient, or ambient ring">ambient</a> -- ambient free module of a subquotient, or ambient ring</span></li>
<li><span><a href="_generators.html" title="provide matrix or list of generators">generators</a> -- provide matrix or list of generators</span></li>
<li><span><a href="_relations.html" title="the defining relations">relations</a> -- the defining relations</span></li>
<li><span><a href="_prune.html" title="prune, e.g., compute a minimal presentation">prune</a> -- prune, e.g., compute a minimal presentation</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>subquotient</tt> :</h2>
<ul><li>subquotient(Matrix,Matrix)</li>
<li>subquotient(Matrix,Nothing)</li>
<li>subquotient(Module,Matrix,Matrix)</li>
<li>subquotient(Module,Matrix,Nothing)</li>
<li>subquotient(Module,Nothing,Matrix)</li>
<li>subquotient(Module,Nothing,Nothing)</li>
<li>subquotient(Nothing,Matrix)</li>
</ul>
</div>
</div>
</body>
</html>
