<?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>inducedMap(Module,Module,Matrix) -- compute the induced map</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_induces__Well__Defined__Map.html">next</a> | <a href="_induced__Map_lp__Module_cm__Module_rp.html">previous</a> | <a href="_induces__Well__Defined__Map.html">forward</a> | <a href="_induced__Map_lp__Module_cm__Module_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>inducedMap(Module,Module,Matrix) -- compute the induced map</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>inducedMap(M,N)</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_induced__Map.html" title="compute an induced map">inducedMap</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
<li><span><tt>N</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
<li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a homomorphism <tt>P <-- Q</tt></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the homomorphism <tt>M <-- N</tt> induced by <tt>f</tt>.</span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_induced__Map_lp..._cm_sp__Degree_sp_eq_gt_sp..._rp.html">Degree => ...</a>, -- specify the degree of a map</span></li>
<li><span><a href="_induced__Map_lp..._cm_sp__Verify_sp_eq_gt_sp..._rp.html">Verify => ...</a>, </span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>The modules <tt>M</tt> and <tt>N</tt> must both be <a href="_subquotient_spmodules.html" title="the way Macaulay2 represents modules">subquotient modules</a> where M and P have the same ambient module, and N and Q have the same ambient module. If the optional argument <tt>Verify</tt> is given, check that the result defines a well defined homomorphism.<p/>
In this example, the module K2 is mapped via g into K1, and we construct the induced map from K2 to K1.<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[x,y,z]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : g1 = matrix{{x,y,z}}
o2 = | x y z |
1 3
o2 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i3 : g2 = matrix{{x^2,y^2,z^2}}
o3 = | x2 y2 z2 |
1 3
o3 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i4 : K1 = ker g1
o4 = image {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3
o4 : R-module, submodule of R</pre>
</td></tr>
<tr><td><pre>i5 : K2 = ker g2
o5 = image {2} | -y2 0 -z2 |
{2} | x2 -z2 0 |
{2} | 0 y2 x2 |
3
o5 : R-module, submodule of R</pre>
</td></tr>
<tr><td><pre>i6 : f = map(ambient K1, ambient K2, {{x,0,0},{0,y,0},{0,0,z}})
o6 = {1} | x 0 0 |
{1} | 0 y 0 |
{1} | 0 0 z |
3 3
o6 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i7 : h = inducedMap(K1,K2,f)
o7 = {2} | xy 0 0 |
{2} | 0 yz 0 |
{2} | 0 0 xz |
o7 : Matrix</pre>
</td></tr>
</table>
If we omit the first argument, then it is understood to be the target of f, and if we omit the second argument, it is understood to be the source of f.<table class="examples"><tr><td><pre>i8 : h1 = inducedMap(target f,K2,f)
o8 = {1} | -xy2 0 -xz2 |
{1} | x2y -yz2 0 |
{1} | 0 y2z x2z |
o8 : Matrix</pre>
</td></tr>
<tr><td><pre>i9 : h2 = inducedMap(,K2,f)
o9 = {1} | -xy2 0 -xz2 |
{1} | x2y -yz2 0 |
{1} | 0 y2z x2z |
o9 : Matrix</pre>
</td></tr>
<tr><td><pre>i10 : h1 == h2
o10 = true</pre>
</td></tr>
</table>
In this example, we cannot omit the second argument, since in that case the resulting object is not a homomorphism.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_induces__Well__Defined__Map.html" title="whether a map is well defined">inducesWellDefinedMap</a> -- whether a map is well defined</span></li>
<li><span><a href="_subquotient.html" title="make a subquotient module">subquotient</a> -- make a subquotient module</span></li>
</ul>
</div>
</div>
</body>
</html>
