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<head><title>cone(ChainComplexMap) -- mapping cone of a chain map</title>
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<div><h1>cone(ChainComplexMap) -- mapping cone of a chain 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>cone f</tt></div>
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<li><span>Function: <a href="_cone.html" title="mapping cone of a chain map">cone</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Chain__Complex__Map.html">chain complex map</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Chain__Complex.html">chain complex</a></span>, the mapping cone of a <tt>f</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y,z]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : m = image vars R
o2 = image | x y z |
1
o2 : R-module, submodule of R</pre>
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<tr><td><pre>i3 : m2 = image symmetricPower(2,vars R)
o3 = image | x2 xy xz y2 yz z2 |
1
o3 : R-module, submodule of R</pre>
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<tr><td><pre>i4 : M = R^1/m2
o4 = cokernel | x2 xy xz y2 yz z2 |
1
o4 : R-module, quotient of R</pre>
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<tr><td><pre>i5 : N = R^1/m
o5 = cokernel | x y z |
1
o5 : R-module, quotient of R</pre>
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<tr><td><pre>i6 : C = cone extend(resolution N,resolution M,id_(R^1))
1 4 9 9 3
o6 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o6 : ChainComplex</pre>
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Let's check that the homology is correct; for example, <tt>HH_0</tt> should be zero.<table class="examples"><tr><td><pre>i7 : prune HH_0 C
o7 = 0
o7 : R-module</pre>
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Let's check that <tt>HH_1</tt> is isomorphic to <tt>m/m2</tt>.<table class="examples"><tr><td><pre>i8 : prune HH_1 C
o8 = cokernel {1} | z y x 0 0 0 0 0 0 |
{1} | 0 0 0 z y x 0 0 0 |
{1} | 0 0 0 0 0 0 z y x |
3
o8 : R-module, quotient of R</pre>
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<tr><td><pre>i9 : prune (m/m2)
o9 = cokernel {1} | z y x 0 0 0 0 0 0 |
{1} | 0 0 0 z y x 0 0 0 |
{1} | 0 0 0 0 0 0 z y x |
3
o9 : R-module, quotient of R</pre>
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