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<head><title>coimage -- coimage of a map</title>
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<div><h1>coimage -- coimage of a 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>coimage f</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, a <a href="___Ring__Map.html">ring map</a>, a <a href="___Matrix.html">matrix</a>, a <a href="___Chain__Complex__Map.html">chain complex map</a>, or a <a href="___Graded__Module__Map.html">graded module map, from A to B</a></span></li>
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<li><div class="single">Outputs:<ul><li><span>The object <tt>A/(kernel f)</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>In each of these cases, the coimage is isomorphic to the image, but the coimage is presented as a quotient object of the source of the map, whereas the image is presented as a subobject of the target of the map. For rings, we can represent quotient rings, but not subrings, in Macaulay2. Hence, for ring maps, we can compute the coimage, but not the image.<p>The isomorphism between coimage and image is not always obvious, as the following example shows.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : M = matrix{{a^3,b^3-c^3,a*b*c,a*(b^2-c^2)}}
o2 = | a3 b3-c3 abc ab2-ac2 |
1 4
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : image M
o3 = image | a3 b3-c3 abc ab2-ac2 |
1
o3 : R-module, submodule of R</pre>
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<tr><td><pre>i4 : coimage M
o4 = cokernel {3} | 0 -bc 0 -b2+c2 |
{3} | a 0 0 0 |
{3} | b-c a2 b2-c2 0 |
{3} | -b-c 0 -bc a2 |
4
o4 : R-module, quotient of R</pre>
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<tr><td><pre>i5 : kernel M
o5 = image {3} | 0 -bc 0 -b2+c2 |
{3} | a 0 0 0 |
{3} | b-c a2 b2-c2 0 |
{3} | -b-c 0 -bc a2 |
4
o5 : R-module, submodule of R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_image.html" title="image of a map">image</a> -- image of a map</span></li>
<li><span><a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel</a> -- cokernel of a map of modules, graded modules, or chaincomplexes</span></li>
<li><span><a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a> -- kernel of a ringmap, matrix, or chain complex</span></li>
<li><span><a href="_comodule.html" title="submodule to quotient module">comodule</a> -- submodule to quotient module</span></li>
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<div class="waystouse"><h2>Ways to use <tt>coimage</tt> :</h2>
<ul><li>coimage(ChainComplexMap)</li>
<li>coimage(GradedModuleMap)</li>
<li>coimage(Matrix)</li>
<li>coimage(RingMap)</li>
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