<?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>localResolution -- find a resolution over a local ring</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_localsyz.html">next</a> | <a href="_local__Prune.html">previous</a> | <a href="_localsyz.html">forward</a> | <a href="_local__Prune.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>localResolution -- find a resolution over a local ring</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>localResolution M</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span></span></li>
<li><span> or </span></li>
<li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></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></span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>DegreeLimit => ...</tt> (missing documentation<!-- tag: localResolution(..., DegreeLimit => ...) -->), </span></li>
<li><span><tt>HardDegreeLimit => ...</tt> (missing documentation<!-- tag: localResolution(..., HardDegreeLimit => ...) -->), </span></li>
<li><span><tt>LengthLimit => ...</tt> (missing documentation<!-- tag: localResolution(..., LengthLimit => ...) -->), </span></li>
<li><span><tt>PairLimit => ...</tt> (missing documentation<!-- tag: localResolution(..., PairLimit => ...) -->), </span></li>
<li><span><tt>SortStrategy => ...</tt> (missing documentation<!-- tag: localResolution(..., SortStrategy => ...) -->), </span></li>
<li><span><tt>StopBeforeComputation => ...</tt> (missing documentation<!-- tag: localResolution(..., StopBeforeComputation => ...) -->), </span></li>
<li><span><tt>Strategy => ...</tt> (missing documentation<!-- tag: localResolution(..., Strategy => ...) -->), </span></li>
<li><span><tt>SyzygyLimit => ...</tt> (missing documentation<!-- tag: localResolution(..., SyzygyLimit => ...) -->), </span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>This method has option inputs that it inherits from </p>
<a href="../../Macaulay2Doc/html/_resolution.html" title="projective resolution">resolution</a>.<p>This function iterates </p>
<a href="_localsyz.html" title="find syzygies">localsyz</a> to obtain a resolution over the local ring.<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[x,y,z,w,SkewCommutative=>true]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : m = matrix{{x,y*z},{z*w,x}}
o2 = | x yz |
| zw x |
2 2
o2 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i3 : setMaxIdeal(ideal(x,y,z,w))
o3 = ideal (x, y, z, w)
o3 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i4 : C = localResolution(coker m, LengthLimit=>10)
2 2 2 2 2 2 2 2 2 2 2
o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5 6 7 8 9 10
o4 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i5 : C = localResolution(coker m)
2 2 2 2 2 2 2 2 2 2 2
o5 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5 6 7 8 9 10
o5 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i6 : C^2
o6 = 0
o6 : R-module</pre>
</td></tr>
<tr><td><pre>i7 : C.dd_4
o7 = {6} | -zw x |
{6} | -x yz |
2 2
o7 : Matrix R <--- R</pre>
</td></tr>
</table>
<table class="examples"><tr><td><pre>i8 : R = QQ[x,y,z]
o8 = R
o8 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i9 : setMaxIdeal ideal vars R
o9 = ideal (x, y, z)
o9 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i10 : m = matrix {{x-1, y, z-1}}
o10 = | x-1 y z-1 |
1 3
o10 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i11 : C = resolution coker m
1 3 3 1
o11 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o11 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i12 : C.dd
1 3
o12 = 0 : R <----------------- R : 1
| x-1 y z-1 |
3 3
1 : R <------------------------- R : 2
{1} | -y -z+1 0 |
{1} | x-1 0 -z+1 |
{1} | 0 x-1 y |
3 1
2 : R <--------------- R : 3
{2} | z-1 |
{2} | -y |
{2} | x-1 |
1
3 : R <----- 0 : 4
0
o12 : ChainComplexMap</pre>
</td></tr>
<tr><td><pre>i13 : LC = localResolution coker m
1 3 2
o13 = R <-- R <-- R <-- 0
0 1 2 3
o13 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i14 : LC.dd
1 3
o14 = 0 : R <----------------- R : 1
| z-1 y x-1 |
3 2
1 : R <--------------------- R : 2
{1} | -x+1 y |
{1} | 0 -z+1 |
{1} | z-1 0 |
2
2 : R <----- 0 : 3
0
o14 : ChainComplexMap</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_set__Max__Ideal.html" title="set the maximal ideal for local ring methods">setMaxIdeal</a> -- set the maximal ideal for local ring methods</span></li>
<li><span><a href="_local__Complement.html" title="find the splitting of the target of a map">localComplement</a> -- find the splitting of the target of a map</span></li>
<li><span><a href="_localsyz.html" title="find syzygies">localsyz</a> -- find syzygies</span></li>
<li><span><a href="_local__Mingens.html" title="finds a minimal set of generators">localMingens</a> -- finds a minimal set of generators</span></li>
<li><span><a href="_local__Modulo.html" title="find the pre-image (pullback) of image of a map over a local ring">localModulo</a> -- find the pre-image (pullback) of image of a map over a local ring</span></li>
<li><span><a href="_local__Prune.html" title="find a minimal presentation">localPrune</a> -- find a minimal presentation</span></li>
<li><span><tt>residueMap</tt> (missing documentation<!-- tag: residueMap -->)</span></li>
<li><span><tt>maxIdeal</tt> (missing documentation<!-- tag: maxIdeal -->)</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>localResolution</tt> :</h2>
<ul><li>localResolution(Ideal)</li>
<li>localResolution(Module)</li>
</ul>
</div>
</div>
</body>
</html>
