<?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>makeS2 -- compute the S2ification of a reduced ring</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Radical__Codim1.html">next</a> | <a href="___Keep.html">previous</a> | <a href="___Radical__Codim1.html">forward</a> | <a href="___Keep.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>makeS2 -- compute the S2ification of a reduced 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>(F,G) = makeS2 R</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, an equidimensional reduced (or just unmixed and genericaly Gorenstein) affine ring</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>F</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span>, <i>R →S</i>, where <i>S</i> is the so-called S2-ification of <i>R</i></span></li>
<li><span><tt>G</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span>, <i>frac S →frac R</i>, giving the corresponding fractions</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>A ring <i>S</i> satisfies Serre’s S2 condition if every codimension 1 ideal contains a nonzerodivisor and every principal ideal generated by a nonzerodivisor is equidimensional of codimension one. If <i>R</i> is an affine reduced ring, then there is a unique smallest extension <i>R⊂S⊂frac(R)</i> satisfying S2, and <i>S</i> is finite as an <i>R</i>-module.</p>
<p>There are several methods to compute <i>S</i>. Currently, only two of these methods is implemented in this package. Stay tuned, or help write the other methods!</p>
<div>One simple example is the rational quartic curve.</div>
<table class="examples"><tr><td><pre>i1 : A = ZZ/101[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : I = monomialCurveIdeal(A,{1,3,4})
3 2 2 2 3 2
o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o2 : Ideal of A</pre>
</td></tr>
<tr><td><pre>i3 : R = A/I;</pre>
</td></tr>
<tr><td><pre>i4 : (F,G) = makeS2 R
ZZ
---[w , a, b, c, d]
101 0,0
o4 = (map(-------------------------------------------------------------------
2 2 2
(b*c - a*d, w d - c , w c - b*d, w b - a*c, w a - b , w
0,0 0,0 0,0 0,0 0,0
------------------------------------------------------------------------
------,R,{a, b, c, d}),
- a*d)
------------------------------------------------------------------------
/ ZZ
| ---[w , a, b, c, d]
| 101 0,0
map(frac(R),frac|-------------------------------------------------------
| 2
|(b*c - a*d, w d - c , w c - b*d, w b - a*c, w a
\ 0,0 0,0 0,0 0,0
------------------------------------------------------------------------
\
|
| b*d
------------------|,{---, a, b, c, d}))
2 2 | c
- b , w - a*d)|
0,0 /
o4 : Sequence</pre>
</td></tr>
</table>
<div>This function is probabilistic, and can fail if <i>R</i> is not a domain. If it fails and the characteristic is not too small, then simply rerun it. If the characteristic is small, then another method may be needed. This might mean that you need to either write it or ask us to do so!</div>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div><div>The return value of this function is likely to change in the future</div>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><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>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>makeS2</tt> :</h2>
<ul><li>makeS2(Ring)</li>
</ul>
</div>
</div>
</body>
</html>
