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<head><title>IntegralClosure -- routines for integral closure of affine domains and ideals</title>
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<div><h1>IntegralClosure -- routines for integral closure of affine domains and ideals</h1>
<div class="single"><h2>Description</h2>
<div><p>This package contains several algorithms for computing the integral closure (i.e. normalization) of an affine domain, and also of an ideal.</p>
<div>The basic use of this package is shown in the following example.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]/(x^3-x^2*z^5-z^2*y^5)
o1 = R
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : S = integralClosure R
o2 = S
o2 : QuotientRing</pre>
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<div>Use <a href="_ic__Fractions.html" title="fractions integral over an affine domain">icFractions</a> to see what fractions have been added.</div>
<table class="examples"><tr><td><pre>i3 : icFractions R
3
y z
o3 = {---, x, y, z}
x
o3 : List</pre>
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<div>Look at the ideal of S or the generators of S to see the structure of the integral closure.</div>
<table class="examples"><tr><td><pre>i4 : gens S
o4 = {w , x, y, z}
1,0
o4 : List</pre>
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<tr><td><pre>i5 : trim ideal S
3 2 5 2 2 5
o5 = ideal (w x - y z, w y z + x*z - x , w + y*z - x*y)
1,0 1,0 1,0
o5 : Ideal of QQ[w , x, y, z]
1,0</pre>
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<p>If the characteristic is positive, yet small compared to the degree, but the fraction ring is still separable over a subring, then use <a href="_ic__Frac__P.html" title="compute the integral closure in prime characteristic">icFracP</a>, which is an implementation of an algorithm due to Leonard-Pellikaan, and modified by Singh-Swanson (see arXiv:0901.0871). However, the interface to this routine is likely to change in future releases to more closely match the functions above.</p>
<div>The method used by integralClosure is a modification of the basic algorithm explained in Theo De Jong’s paper <em>An Algorithm for Computing the Integral Closure</em>, J. Symbolic Computation, (1998) 26, 273-277.</div>
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<div class="single"><h2>Authors</h2>
<ul><li><div class="single"><a href="http://faculty1.coloradocollege.edu/~ataylor/">Amelia Taylor</a><span> <<a href="mailto:amelia.taylor@coloradocollege.edu">amelia.taylor@coloradocollege.edu</a>></span></div>
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<li><div class="single"><a href="http://www.msri.org/~de/">David Eisenbud</a><span> <<a href="mailto:de@msri.org">de@msri.org</a>></span></div>
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<li><div class="single"><a href="http://www.math.cornell.edu/~mike">Mike Stillman</a><span> <<a href="mailto:mike@math.cornell.edu">mike@math.cornell.edu</a>></span></div>
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<div class="single"><h2>Version</h2>
This documentation describes version <b>1.0</b> of IntegralClosure.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/IntegralClosure.m2">IntegralClosure.m2</a>. The auxiliary files accompanying it are in the directory <a href="../../../../Macaulay2/IntegralClosure/">IntegralClosure/</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Functions<ul><li><span><a href="_conductor.html" title="the conductor of a finite ring map">conductor</a> -- the conductor of a finite ring map</span></li>
<li><span><a href="_ic__Frac__P.html" title="compute the integral closure in prime characteristic">icFracP</a> -- compute the integral closure in prime characteristic</span></li>
<li><span><a href="_ic__Fractions.html" title="fractions integral over an affine domain">icFractions</a> -- fractions integral over an affine domain</span></li>
<li><span><a href="_ic__Map.html" title="natural map from an affine domain into its integral closure">icMap</a> -- natural map from an affine domain into its integral closure</span></li>
<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="_idealizer.html" title="compute Hom(I,I) as a quotient ring">idealizer</a> -- compute Hom(I,I) as a quotient ring</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>
<li><span><a href="_make__S2.html" title="compute the S2ification of a reduced ring">makeS2</a> -- compute the S2ification of a reduced ring</span></li>
<li><span><a href="_ring__From__Fractions.html" title="find presentation for f.g. ring">ringFromFractions</a> -- find presentation for f.g. ring</span></li>
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<li><div class="single">Symbols<ul><li><span><tt>AllCodimensions</tt> (missing documentation<!-- tag: AllCodimensions -->)</span></li>
<li><span><a href="___Conductor__Element.html" title="Specifies a particular non-zerodivisor in the conductor.">ConductorElement</a> -- Specifies a particular non-zerodivisor in the conductor.</span></li>
<li><span><a href="___Index.html" title="Optional input for idealizer">Index</a> -- Optional input for idealizer</span></li>
<li><span><a href="___Keep.html" title="an optional argument for various functions">Keep</a> -- an optional argument for various functions</span></li>
<li><span><a href="___Radical__Codim1.html" title="a symbol denoting a strategy element usable with integralClosure(...,Strategy=>...)">RadicalCodim1</a> -- a symbol denoting a strategy element usable with integralClosure(...,Strategy=>...)</span></li>
<li><span><a href="___Verbosity.html" title="optional argument describing how verbose the output should be">Verbosity</a> -- optional argument describing how verbose the output should be</span></li>
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