<?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>evansGriffith -- Reduces the rank of a syzygy</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_is__Syzygy.html">next</a> | <a href="_elementary.html">previous</a> | <a href="_is__Syzygy.html">forward</a> | <a href="_elementary.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>evansGriffith -- Reduces the rank of a syzygy</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>N = evansGriffith(M,d)</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, over a polynomial ring whose cokernel is an d-th syzygy.</span></li>
<li><span><tt>d</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, positive</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>N</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with the same source and kernel as M, but such that coker N is a dth syzygy of rank d.</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>The routine reduces the target of M by elementary moves (see <a href="_elementary.html" title="Elementary moves are used to reduce the target of a syzygy matrix">elementary</a>) involving just d+1 variables. The outcome is probabalistic, but if the routine fails, it gives an error message.</p>
<div>See the book of Evans and Griffith (<i>Syzygies.</i> London Mathematical Society Lecture Note Series, 106. Cambridge University Press, Cambridge, 1985.)</div>
<table class="examples"><tr><td><pre>i1 : kk=ZZ/32003
o1 = kk
o1 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i2 : S=kk[a..e]
o2 = S
o2 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i3 : i=ideal(a^2,b^3,c^4, d^5)
2 3 4 5
o3 = ideal (a , b , c , d )
o3 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i4 : F=res i
1 4 6 4 1
o4 = S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5
o4 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i5 : f=F.dd_3
o5 = {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | a2 0 0 d5 |
{7} | 0 -b3 -c4 0 |
{8} | 0 a2 0 -c4 |
{9} | 0 0 a2 b3 |
6 4
o5 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i6 : EG = evansGriffith(f,2) -- notice that we have a matrix with one less row, as described in elementary, and the target module rank is one less.
o6 = {5} | c4 d5 0
{6} | -b3 0 d5
{7} | 0 -b3 7274a4-2219a3b+1698a2b2-3867a3c+15593a2bc+194a2c2-c4
{7} | a2 0 2053a4+10346a3b+4507a2b2-11175a3c+13708a2bc+7857a2c2
{8} | 0 a2 15111a3-7443a2b+6705a2c
------------------------------------------------------------------------
0 |
0 |
7274a2b3-2219ab4+1698b5-3867ab3c+15593b4c+194b3c2 |
2053a2b3+10346ab4+4507b5-11175ab3c+13708b4c+7857b3c2+d5 |
15111ab3-7443b4+6705b3c-c4 |
5 4
o6 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i7 : isSyzygy(coker EG,2)
o7 = true</pre>
</td></tr>
</table>
<div>This is called within <a href="_bruns.html" title="Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module">bruns</a>.</div>
</div>
</div>
<div class="waystouse"><h2>Ways to use <tt>evansGriffith</tt> :</h2>
<ul><li>evansGriffith(Matrix,ZZ)</li>
</ul>
</div>
</div>
</body>
</html>
