<?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>presentation(Module) -- presentation of a module</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_presentation_lp__Polynomial__Ring_cm__Quotient__Ring_rp.html">next</a> | <a href="_presentation.html">previous</a> | <a href="_presentation_lp__Polynomial__Ring_cm__Quotient__Ring_rp.html">forward</a> | <a href="_presentation.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>presentation(Module) -- presentation of a 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>presentation M</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="_presentation.html" title="presentation of a module or ring">presentation</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> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, a presentation matrix of <tt>M</tt></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>A presentation of <tt>M</tt> is a map <tt>p</tt> so that <tt>coker p</tt> is isomorphic to <tt>M</tt>. The presentation obtained is expressed in terms of the given generators, i.e., the modules <tt>cover M</tt> and <tt>target p</tt> are identical. The isomorphism can be obtained as <tt>map(M,coker p,1)</tt>.<p/> Since a module M may be described as a submodule or a subquotient module of a free module, some computation may be required to produce a presentation. See also <a href="_trim.html" title="minimize generators and relations">trim</a>, or <a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a>, which do a bit more work to try to eliminate redundant generators.<table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c];</pre> </td></tr> <tr><td><pre>i2 : I = ideal"a2-b2,abc" 2 2 o2 = ideal (a - b , a*b*c) o2 : Ideal of R</pre> </td></tr> <tr><td><pre>i3 : M = I/(I^2+a*I) o3 = subquotient (| a2-b2 abc |, | a4-2a2b2+b4 a3bc-ab3c a2b2c2 a3-ab2 a2bc |) 1 o3 : R-module, subquotient of R</pre> </td></tr> <tr><td><pre>i4 : presentation M o4 = {2} | a b2 0 0 | {3} | 0 0 a b2 | 2 4 o4 : Matrix R <--- R</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> -- compute a minimal presentation</span></li> <li><span><a href="_trim.html" title="minimize generators and relations">trim</a> -- minimize generators and relations</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="_cover.html" title="get the covering free module">cover</a> -- get the covering free module</span></li> </ul> </div> </div> </body> </html>