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<head><title>cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes</title>
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<div><h1>cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes</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>cokernel f</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f : A --> B</tt><span>a <a href="___Matrix.html">matrix</a></span>, <span>a <a href="___Chain__Complex__Map.html">chain complex map</a></span>, <span>a <a href="___Ring__Element.html">ring element</a></span>, or <span>a <a href="___Graded__Module__Map.html">graded module map</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span>the object <tt>B/(image f)</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><p><tt>coker</tt> is a synonym for <tt>cokernel</tt>.</p>
<p>The generators of the cokernel are provided by the generators of the target of <tt>f</tt>. In other words, <tt>cover target f</tt> and <tt>cover cokernel f</tt> are equal.</p>
<p>An argument <tt>f</tt> that is a <a href="___Ring__Element.html" title="the class of all ring elements handled by the engine">RingElement</a> is interpreted as a one by one matrix.</p>
<table class="examples"><tr><td><pre>i1 : R = ZZ[a..d];</pre>
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<tr><td><pre>i2 : M = cokernel matrix{{2*a-b,3*c-5*d,a^2-b-3}}
o2 = cokernel | 2a-b 3c-5d a2-b-3 |
1
o2 : R-module, quotient of R</pre>
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If <tt>f</tt> is a matrix, and the target of <tt>f</tt> is a submodule, the resulting module will be a <a href="_subquotient.html" title="make a subquotient module">subquotient</a> module.<table class="examples"><tr><td><pre>i3 : f = map(a*M, M, a^3+a^2*b)
o3 = {1} | a+10b+18 |
o3 : Matrix</pre>
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<tr><td><pre>i4 : (target f,source f)
o4 = (subquotient (| a |, | 2a-b 3c-5d a2-b-3 |), cokernel | 2a-b 3c-5d
------------------------------------------------------------------------
a2-b-3 |)
o4 : Sequence</pre>
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<tr><td><pre>i5 : N = cokernel f
o5 = subquotient (| a |, | a2+10ab+18a 2a-b 3c-5d a2-b-3 |)
1
o5 : R-module, subquotient of R</pre>
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<tr><td><pre>i6 : minimalPresentation N
o6 = cokernel | 3c-5d 3b-18 21b+36 a+10b+18 b2-4b-12 |
1
o6 : R-module, quotient of R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_cover.html" title="get the covering free module">cover</a> -- get the covering free module</span></li>
<li><span><a href="_image.html" title="image of a map">image</a> -- image of a map</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="_coimage.html" title="coimage of a map">coimage</a> -- coimage of a map</span></li>
<li><span><a href="_comodule.html" title="submodule to quotient module">comodule</a> -- submodule to quotient module</span></li>
<li><span><a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> -- compute a minimal presentation</span></li>
<li><span><a href="_matrices_spto_spand_spfrom_spmodules.html" title="including kernel, cokernel and image">matrices to and from modules</a> -- including kernel, cokernel and image</span></li>
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<div class="waystouse"><h2>Ways to use <tt>cokernel</tt> :</h2>
<ul><li>cokernel(ChainComplexMap)</li>
<li>cokernel(GradedModuleMap)</li>
<li>cokernel(Matrix)</li>
<li>cokernel(RingElement)</li>
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