<?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>Module ** Module -- tensor product</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Module_sp_st_st_sp__Ring.html">next</a> | <a href="_module.html">previous</a> | <a href="___Module_sp_st_st_sp__Ring.html">forward</a> | <a href="_module.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>Module ** Module -- tensor product</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>M ** N</tt></div>
</dd></dl>
</div>
</li>
<li><span>Operator: <a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</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>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, the tensor product of M and N</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>If M has generators m1, m2, ..., mr, and N has generators n1, n2, ..., ns, then M ** N has generators: m1**n1, m1**n2, ..., m2**n1, ..., mr**ns.<table class="examples"><tr><td><pre>i1 : R = ZZ[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : M = image matrix {{a,b}}
o2 = image | a b |
1
o2 : R-module, submodule of R</pre>
</td></tr>
<tr><td><pre>i3 : N = image matrix {{c,d}}
o3 = image | c d |
1
o3 : R-module, submodule of R</pre>
</td></tr>
<tr><td><pre>i4 : M ** N
o4 = cokernel {2} | -d 0 -b 0 |
{2} | c 0 0 -b |
{2} | 0 -d a 0 |
{2} | 0 c 0 a |
4
o4 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i5 : N ** M
o5 = cokernel {2} | -b 0 -d 0 |
{2} | a 0 0 -d |
{2} | 0 -b c 0 |
{2} | 0 a 0 c |
4
o5 : R-module, quotient of R</pre>
</td></tr>
</table>
<p/>
Use <a href="_trim.html" title="minimize generators and relations">trim</a> or <a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> if a more compact presentation is desired.<p/>
Use <a href="_flip_lp__Module_cm__Module_rp.html" title="matrix of commutativity of tensor product">flip(Module,Module)</a> to produce the isomorphism M ** N --> N ** M.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_flip_lp__Module_cm__Module_rp.html" title="matrix of commutativity of tensor product">flip</a> -- matrix of commutativity of tensor product</span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Module.html" title="tensor product">Module ** Matrix</a> -- tensor product</span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Matrix.html" title="tensor product">Matrix ** Matrix</a> -- tensor product</span></li>
</ul>
</div>
</div>
</body>
</html>
