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<head><title>icMap -- natural map from an affine domain into its integral closure</title>
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<div><h1>icMap -- natural map from an affine domain into its integral closure</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>icMap R</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, an affine domain</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span>, from <tt>R</tt> to its integral closure</span></li>
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<div class="single"><h2>Description</h2>
<div><div>If the integral closure of <tt>R</tt> has not yet been computed, that computation is performed first. No extra computation is involved. If <tt>R</tt> is integrally closed, then the identity map is returned.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y]/(y^2-x^3)
o1 = R
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : icMap R
QQ[w , x, y]
0,0
o2 = map(---------------------------------,R,{x, y})
2 2
(w y - x , w x - y, w - x)
0,0 0,0 0,0
QQ[w , x, y]
0,0
o2 : RingMap --------------------------------- <--- R
2 2
(w y - x , w x - y, w - x)
0,0 0,0 0,0</pre>
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<p></p>
<div>This finite ring map can be used to compute the conductor, that is, the ideal of elements of <tt>R</tt> which are universal denominators for the integral closure (i.e. those d ∈R such that d R’ ⊂R).</div>
<table class="examples"><tr><td><pre>i3 : S = QQ[a,b,c]/ideal(a^6-c^6-b^2*c^4);</pre>
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<tr><td><pre>i4 : F = icMap S;
QQ[w , w , a, b, c]
4,0 3,0
o4 : RingMap --------------------------------------------------------- <--- S
2 2 2 2 2
(w c - a , w c - w a, w a - w , w - b - c )
3,0 4,0 3,0 4,0 3,0 4,0</pre>
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<tr><td><pre>i5 : conductor F
3 2 3 4
o5 = ideal (c , a*c , a c, a )
o5 : Ideal of S</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>If you want to control the computation of the integral closure via optional arguments, then make sure you call <a href="_integral__Closure_lp__Ring_rp.html" title="compute the integral closure (normalization) of an affine domain">integralClosure(Ring)</a> first, since <tt>icMap</tt> does not have optional arguments.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_integral__Closure_lp__Ring_rp.html" title="compute the integral closure (normalization) of an affine domain">integralClosure(Ring)</a> -- compute the integral closure (normalization) of an affine domain</span></li>
<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>
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<div class="waystouse"><h2>Ways to use <tt>icMap</tt> :</h2>
<ul><li>icMap(Ring)</li>
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