<?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>Singular Book 2.1.13 -- kernel, image and cokernel of a module homomorphism</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Singular_sp__Book_sp2.1.20.html">next</a> | <a href="___Singular_sp__Book_sp2.1.10.html">previous</a> | <a href="___Singular_sp__Book_sp2.1.20.html">forward</a> | <a href="___Singular_sp__Book_sp2.1.10.html">backward</a> | <a href="___M2__Singular__Book.html">up</a> | <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>
<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp2.1.13.html" title="kernel, image and cokernel of a module homomorphism">Singular Book 2.1.13</a></div>
<hr/>
<div><h1>Singular Book 2.1.13 -- kernel, image and cokernel of a module homomorphism</h1>
<div>In Macaulay2, a Matrix is the same thing as a module homomorphism. The computation of the kernel of a module homomorphism is based on a Groebner basis computation.<table class="examples"><tr><td><pre>i1 : A = QQ[x,y,z];</pre>
</td></tr>
<tr><td><pre>i2 : M = matrix{{x,x*y,z},{x^2,x*y*z,y*z}}
o2 = | x xy z |
| x2 xyz yz |
2 3
o2 : Matrix A <--- A</pre>
</td></tr>
<tr><td><pre>i3 : K = kernel M
o3 = image {2} | -y2z+yz2 |
{3} | -xz+yz |
{2} | x2y-xyz |
3
o3 : A-module, submodule of A</pre>
</td></tr>
</table>
The image and cokernel of a matrix require no computation.<table class="examples"><tr><td><pre>i4 : I = image M
o4 = image | x xy z |
| x2 xyz yz |
2
o4 : A-module, submodule of A</pre>
</td></tr>
<tr><td><pre>i5 : N = cokernel M
o5 = cokernel | x xy z |
| x2 xyz yz |
2
o5 : A-module, quotient of A</pre>
</td></tr>
<tr><td><pre>i6 : P = coimage M
o6 = cokernel {2} | -y2z+yz2 |
{3} | -xz+yz |
{2} | x2y-xyz |
3
o6 : A-module, quotient of A</pre>
</td></tr>
</table>
</div>
</div>
</body>
</html>
