<?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>icFracP -- compute the integral closure in prime characteristic</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_ic__Frac__P_lp..._cm_sp__Conductor__Element_sp_eq_gt_sp..._rp.html">next</a> | <a href="___Conductor__Element.html">previous</a> | <a href="_ic__Frac__P_lp..._cm_sp__Conductor__Element_sp_eq_gt_sp..._rp.html">forward</a> | <a href="___Conductor__Element.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>icFracP -- compute the integral closure in prime characteristic</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>icFracP R, icFracP(R, ConductorElement => D), icFracP(R, Limit => N), icFracP(R, Verbosity => ZZ)</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, that is reduced, equidimensional, finitely and separably generated over a field of characteristic p</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span>The module generators of the integral closure of <tt>R</tt> in its total ring of fractions. The generators are given as elements in the total ring of fractions.</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>ConductorElement => </tt><span><span>default value null</span>, optionally provide a non-zerodivisor conductor element <tt>ConductorElement => D</tt>; the output is then the module generators of the integral closure. A good choice of <tt>D</tt> may speed up the calculations?</span></span></li>
<li><span><tt>Limit => </tt><span><span>default value infinity</span>, if value N is given, perform N loop steps only</span></span></li>
<li><span><tt>Verbosity => </tt><span><span>default value 0</span>, if value is greater than 0, report the conductor element and number of steps in the loop</span></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>Input is an equidimensional reduced ring in characteristic p that is finitely and separably generated over the base field. The output is a finite set of fractions that generate the integral closure as an <tt>R</tt>-module. An intermediate step in the code is the computation of a conductor element <tt>D</tt> that is a non-zerodivisor; its existence is guaranteed by the separability assumption. The user may supply <tt>D</tt> with the optional <tt>ConductorElement => D</tt>. (Sometimes, but not always, supplying <tt>D</tt> speeds up the computation.) In any case, with the non-zero divisor <tt>D</tt>, the algorithm starts by setting the initial approximation of the integral closure to be the finitely generated <tt>R</tt>-module <tt>(1/D)R</tt>, and in the subsequent loop the code recursively constructs submodules. Eventually two submodules repeat; the repeated module is the integral closure of <tt>R</tt>. The user may optionally provide <tt>Limit => N</tt> to stop the loop after <tt>N</tt> steps, and the optional <tt>Verbosity => 1</tt> reports the conductor element and the number of steps it took for the loop to stabilize. The algorithm is based on the Leonard--Pellikaan--Singh--Swanson algorithm.<p/>
A simple example.<table class="examples"><tr><td><pre>i1 : R = ZZ/5[x,y,z]/ideal(x^6-z^6-y^2*z^4);</pre>
</td></tr>
<tr><td><pre>i2 : icFracP R
2 3 2 3
x x y z + z
o2 = {1, --, --, --------}
z 2 x
z
o2 : List</pre>
</td></tr>
</table>
The user may provide an optional non-zerodivisor conductor element <tt>D</tt>. The output generators need not be expressed in the form with denominator <tt>D</tt>.<table class="examples"><tr><td><pre>i3 : R = ZZ/5[x,y,u,v]/ideal(x^2*u-y^2*v);</pre>
</td></tr>
<tr><td><pre>i4 : icFracP(R)
x*u
o4 = {1, ---}
y
o4 : List</pre>
</td></tr>
<tr><td><pre>i5 : icFracP(R, ConductorElement => x)
x*u
o5 = {1, ---}
y
o5 : List</pre>
</td></tr>
</table>
In case <tt>D</tt> is not in the conductor, the output is <tt>V_e = (1/D) {r in R | r^(p^i) in (D^(p^i-1)) </tt>for <tt>i = 1, ..., e}</tt> such that <tt>V_e = V_(e+1)</tt> and <tt>e</tt> is the smallest such <tt>e</tt>.<table class="examples"><tr><td><pre>i6 : R=ZZ/2[u,v,w,x,y,z]/ideal(u^2*x^3+u*v*y^3+v^2*z^3);</pre>
</td></tr>
<tr><td><pre>i7 : icFracP(R)
3 3
u*y + v*z
o7 = {1, -----------}
u
o7 : List</pre>
</td></tr>
<tr><td><pre>i8 : icFracP(R, ConductorElement => x^2)
o8 = {1}
o8 : List</pre>
</td></tr>
</table>
The user may also supply an optional limit on the number of steps in the algorithm. In this case, the output is a finitely generated <tt>R</tt>-module contained in <tt>(1/D)R</tt> which contains the integral closure (intersected with <tt>(1/D)R</tt>.<table class="examples"><tr><td><pre>i9 : R=ZZ/2[u,v,w,x,y,z]/ideal(u^2*x^3+u*v*y^3+v^2*z^3);</pre>
</td></tr>
<tr><td><pre>i10 : icFracP(R, Limit => 1)
2
1 u*x
o10 = {-, ----}
y v*y
o10 : List</pre>
</td></tr>
<tr><td><pre>i11 : icFracP(R, Limit => 2)
3 3
u*y + v*z
o11 = {1, -----------}
u
o11 : List</pre>
</td></tr>
<tr><td><pre>i12 : icFracP(R)
3 3
u*y + v*z
o12 = {1, -----------}
u
o12 : List</pre>
</td></tr>
</table>
With the option above one can for example determine how many intermediate modules the program should compute or did compute in the loop to get the integral closure. A shortcut for finding the number of steps performed is to supply the <tt>Verbosity => 1</tt> option.<table class="examples"><tr><td><pre>i13 : R=ZZ/3[u,v,w,x,y,z]/ideal(u^2*x^4+u*v*y^4+v^2*z^4);</pre>
</td></tr>
<tr><td><pre>i14 : icFracP(R, Verbosity => 1)
Number of steps: 4, Conductor Element: u*x^4-v*y^4
4 4
u*y - v*z
o14 = {1, -----------}
u
o14 : List</pre>
</td></tr>
</table>
With this extra bit of information, the user can now compute integral closures of principal ideals in <tt>R</tt> via <a href="_ic__P__Ideal.html" title="compute the integral closure in prime characteristic of a principal ideal">icPIdeal</a>.</div>
</div>
<div class="single"><h2>Caveat</h2>
<div>The interface to this algorithm will likely change in Macaulay2 1.4</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_ic__P__Ideal.html" title="compute the integral closure in prime characteristic of a principal ideal">icPIdeal</a> -- compute the integral closure in prime characteristic of a principal ideal</span></li>
<li><span><a href="_integral__Closure.html" title="integral closure of an ideal or a domain">integralClosure</a> -- integral closure of an ideal or a domain</span></li>
<li><span><a href="_is__Normal.html" title="determine if a reduced ring is normal">isNormal</a> -- determine if a reduced ring is normal</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>icFracP</tt> :</h2>
<ul><li>icFracP(Ring)</li>
</ul>
</div>
</div>
</body>
</html>
