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<head><title>homology(Matrix,Matrix) -- homology of a pair of maps</title>
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<div><h1>homology(Matrix,Matrix) -- homology of a pair of maps</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>M = homology(f,g)</tt></div>
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<li><span>Function: <a href="_homology.html" title="general homology functor">homology</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span></span></li>
<li><span><tt>g</tt>, <span>a <a href="___Matrix.html">matrix</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, computes the homology module <tt>(kernel f)/(image g)</tt>.</span></li>
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<div class="single"><h2>Description</h2>
<div>Here <tt>g</tt> and <tt>f</tt> should be composable maps with <tt>g*f</tt> equal to zero.<p>In the following example, we ensure that the source of <tt>f</tt> and the target of <tt>f</tt> are exactly the same, taking even the degrees into account, and we ensure that <tt>f</tt> is homogeneous.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x]/x^5;</pre>
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<tr><td><pre>i2 : f = map(R^1,R^1,{{x^3}}, Degree => 3)
o2 = | x3 |
1 1
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : M = homology(f,f)
o3 = subquotient (| x2 |, | x3 |)
1
o3 : R-module, subquotient of R</pre>
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<tr><td><pre>i4 : prune M
o4 = cokernel {2} | x |
1
o4 : R-module, quotient of R</pre>
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