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<head><title>conductor -- the conductor of a finite ring map</title>
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<div><h1>conductor -- the conductor of a finite ring map</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>conductor F</tt><br/><tt>conductor R</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span>, <i>R →S</i>, a finite map with <i>R</i> an affine reduced ring</span></li>
<li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, an affine domain. In this case, <i>F : R →S</i> is the inclusion map of <i>R</i> into the integral closure <i>S</i></span></li>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, of <i>R</i> consisting of all <i>d ∈R</i> such that <i>dS ⊂F(R)</i></span></li>
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<div class="single"><h2>Description</h2>
<div><div>Suppose that the ring map <i>F : R →S</i> is finite: i.e. <i>S</i> is a finitely generated <i>R</i>-module. The conductor of <i>F</i> is defined to be <i>{g ∈R | g S ⊂F(R) }</i>. One way to think about this is that the conductor is the set of universal denominators of <tt>S</tt> over <tt>R</tt>, or as the largest ideal of <tt>R</tt> which is also an ideal in <tt>S</tt>. An important case is the conductor of the map from a ring to its integral closure.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]/ideal(x^7-z^7-y^2*z^5);</pre>
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<tr><td><pre>i2 : icFractions R
3 2
x x
o2 = {--, --, x, y, z}
2 z
z
o2 : List</pre>
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<tr><td><pre>i3 : F = icMap R
QQ[w , w , x, y, z]
5,0 4,0
o3 = map(--------------------------------------------------------------------------------------------------------,R,{x, y, z})
2 2 2 2 3 2 2 2 3 2 2
(w z - x , w z - w x, w x - w , w x - y z - z , w w - x*y - x*z , w - w y - x z)
4,0 5,0 4,0 5,0 4,0 5,0 5,0 4,0 5,0 4,0
QQ[w , w , x, y, z]
5,0 4,0
o3 : RingMap -------------------------------------------------------------------------------------------------------- <--- R
2 2 2 2 3 2 2 2 3 2 2
(w z - x , w z - w x, w x - w , w x - y z - z , w w - x*y - x*z , w - w y - x z)
4,0 5,0 4,0 5,0 4,0 5,0 5,0 4,0 5,0 4,0</pre>
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<tr><td><pre>i4 : conductor F
4 3 2 2 4 5
o4 = ideal (z , x*z , x z , x z, x )
o4 : Ideal of R</pre>
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<div>If an affine domain (a ring finitely generated over a field) is given as input, then the conductor of <i>R</i> in its integral closure is returned.</div>
<table class="examples"><tr><td><pre>i5 : conductor R
4 3 2 2 4 5
o5 = ideal (z , x*z , x z , x z, x )
o5 : Ideal of R</pre>
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<p></p>
<div>If the map is not <tt>icFractions(R)</tt>, then <a href="../../Macaulay2Doc/html/_push__Forward_lp__Ring__Map_cm__Module_rp.html" title="">pushForward</a> is called to compute the conductor.</div>
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<div class="single"><h2>Caveat</h2>
<div><div>Currently this function only works if <tt>F</tt> comes from a integral closure computation, or is homogeneous</div>
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<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>
<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="../../Macaulay2Doc/html/_push__Forward_lp__Ring__Map_cm__Module_rp.html" title="">pushForward</a></span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>conductor</tt> :</h2>
<ul><li>conductor(Ring)</li>
<li>conductor(RingMap)</li>
</ul>
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