<?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>tateResolution -- finite piece of the Tate resolution</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div>next | <a href="_sym__Ext.html">previous</a> | forward | <a href="_sym__Ext.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>tateResolution -- finite piece of the Tate resolution</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>tateResolution(m,E,l,h)</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>, a presentation matrix for a module</span></li>
<li><span><tt>E</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, exterior algebra</span></li>
<li><span><tt>l</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, lower cohomological degree</span></li>
<li><span><tt>h</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, upper bound on the cohomological degree</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Chain__Complex.html">chain complex</a></span>, a finite piece of the Tate resolution</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>This function takes as input a presentation matrix <tt>m</tt> of a finitely generated graded <tt>S</tt>-module <tt>M</tt> an exterior algebra <tt>E</tt> and two integers <tt>l</tt> and <tt>h</tt>. If <tt>r</tt> is the regularity of <tt>M</tt>, then this function computes the piece of the Tate resolution from cohomological degree <tt>l</tt> to cohomological degree <tt>max(r+2,h)</tt>. For instance, for the homogeneous coordinate ring of a point in the projective plane:<table class="examples"><tr><td><pre>i1 : S = ZZ/32003[x_0..x_2]; </pre>
</td></tr>
<tr><td><pre>i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];</pre>
</td></tr>
<tr><td><pre>i3 : m = matrix{{x_0,x_1}};
1 2
o3 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i4 : regularity coker m
o4 = 0</pre>
</td></tr>
<tr><td><pre>i5 : T = tateResolution(m,E,-2,4)
1 1 1 1 1 1 1
o5 = E <-- E <-- E <-- E <-- E <-- E <-- E
0 1 2 3 4 5 6
o5 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i6 : betti T
0 1 2 3 4 5 6
o6 = total: 1 1 1 1 1 1 1
-4: 1 1 1 1 1 1 1
o6 : BettiTally</pre>
</td></tr>
<tr><td><pre>i7 : T.dd_1
o7 = {-4} | e_2 |
1 1
o7 : Matrix E <--- E</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_sym__Ext.html" title="the first differential of the complex R(M)">symExt</a> -- the first differential of the complex R(M)</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>tateResolution</tt> :</h2>
<ul><li>tateResolution(Matrix,PolynomialRing,ZZ,ZZ)</li>
</ul>
</div>
</div>
</body>
</html>
