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<head><title>homomorphism -- get the homomorphism from element of Hom</title>
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<div><h1>homomorphism -- get the homomorphism from element of Hom</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>homomorphism f</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, of the form Hom(M,N) <-- R^1, where Hom(M,N) has been previously computed, and R is the ring of f, M and N</span></li>
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<li><div class="single">Outputs:<ul><li><span>the <a href="___Matrix.html" title="the class of all matrices">Matrix</a> <tt>M --> N</tt>, corresponding to the element <tt>f</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>When <tt>H := Hom(M,N)</tt> is computed, enough information is stored in <tt>H.cache.Hom</tt> to compute this correspondence.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]/(y^2-x^3)
o1 = R
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : H = Hom(ideal(x,y), R^1)
o2 = image {-1} | x y |
{-1} | y x2 |
2
o2 : R-module, submodule of R</pre>
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<tr><td><pre>i3 : g = homomorphism H_{1}
o3 = | y x2 |
o3 : Matrix</pre>
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The homomorphism g takes x to y and y to x2. The source and target are what they should be.<table class="examples"><tr><td><pre>i4 : source g
o4 = image | x y |
1
o4 : R-module, submodule of R</pre>
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<tr><td><pre>i5 : target g
1
o5 = R
o5 : R-module, free</pre>
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<p/>
After <a href="_minimal__Presentation_lp__Module_rp.html">pruning</a> a Hom module, one cannot use homomorphism directly. Instead, first apply the pruning map:<table class="examples"><tr><td><pre>i6 : H1 = prune H
o6 = cokernel | x2 -y |
| -y x |
2
o6 : R-module, quotient of R</pre>
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<tr><td><pre>i7 : homomorphism(H1.cache.pruningMap * H1_{1})
o7 = | y x2 |
o7 : Matrix</pre>
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Sometime, one wants a random homomorphism of a given degree. Here is one method:<table class="examples"><tr><td><pre>i8 : f = basis(3,H)
o8 = {0} | xy2 xyz xz2 y3 y2z yz2 z3 0 0 0 |
{1} | 0 0 0 0 0 0 0 y2 yz z2 |
o8 : Matrix</pre>
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<tr><td><pre>i9 : rand = random(R^(numgens source f), R^1)
o9 = | 3 |
| 1 |
| 3 |
| 3/4 |
| 1/5 |
| 2/7 |
| 6/7 |
| 10/3 |
| 4 |
| 7/3 |
10 1
o9 : Matrix R <--- R</pre>
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<tr><td><pre>i10 : h = homomorphism(f * rand)
o10 = | 3x2y2+3/4xy3+x2yz+1/5xy2z+3x2z2+2/7xyz2+6/7xz3+10/3y3+4y2z+7/3yz2
-----------------------------------------------------------------------
10/3x2y2+3xy3+3/4y4+4x2yz+xy2z+1/5y3z+7/3x2z2+3xyz2+2/7y2z2+6/7yz3 |
o10 : Matrix</pre>
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<tr><td><pre>i11 : source h
o11 = image | x y |
1
o11 : R-module, submodule of R</pre>
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<tr><td><pre>i12 : target h
1
o12 = R
o12 : R-module, free</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Hom.html" title="module of homomorphisms">Hom</a> -- module of homomorphisms</span></li>
<li><span><a href="_prune.html" title="prune, e.g., compute a minimal presentation">prune</a> -- prune, e.g., compute a minimal presentation</span></li>
<li><span><a href="_random.html" title="get a random element">random</a> -- get a random element</span></li>
<li><span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>homomorphism</tt> :</h2>
<ul><li>homomorphism(Matrix)</li>
</ul>
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