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<head><title>localModulo -- find the pre-image (pullback) of image of a map over a local ring</title>
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<div><h1>localModulo -- find the pre-image (pullback) of image of a map over a local 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>localModulo(m,n)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>m</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></li>
<li><span><tt>n</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose image is the pre-image (pullback) of the image of n under m</span></li>
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<div class="single"><h2>Description</h2>
<div>The maps m and n must have the same target, and their sources and targets must be free. If m is null, then it is taken to be the identity. If n is null, it is taken to be zero.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : setMaxIdeal ideal vars R
o2 = ideal (x, y, z)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : m = matrix {{x-1, y}}
o3 = | x-1 y |
1 2
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : n = matrix {{y,z}}
o4 = | y z |
1 2
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : modulo (m,n)
o5 = {1} | 0 z 0 y |
{1} | -1 0 z -x+1 |
2 4
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : localModulo (m,n)
o6 = {1} | z y 0 |
{1} | 0 -x+1 -1 |
2 3
o6 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_set__Max__Ideal.html" title="set the maximal ideal for local ring methods">setMaxIdeal</a> -- set the maximal ideal for local ring methods</span></li>
<li><span><a href="_local__Complement.html" title="find the splitting of the target of a map">localComplement</a> -- find the splitting of the target of a map</span></li>
<li><span><a href="_localsyz.html" title="find syzygies">localsyz</a> -- find syzygies</span></li>
<li><span><a href="_local__Mingens.html" title="finds a minimal set of generators">localMingens</a> -- finds a minimal set of generators</span></li>
<li><span><a href="_local__Prune.html" title="find a minimal presentation">localPrune</a> -- find a minimal presentation</span></li>
<li><span><a href="_local__Resolution.html" title="find a resolution over a local ring">localResolution</a> -- find a resolution over a local ring</span></li>
<li><span><tt>residueMap</tt> (missing documentation<!-- tag: residueMap -->)</span></li>
<li><span><tt>maxIdeal</tt> (missing documentation<!-- tag: maxIdeal -->)</span></li>
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<div class="waystouse"><h2>Ways to use <tt>localModulo</tt> :</h2>
<ul><li>localModulo(Matrix,Matrix)</li>
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