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<head><title>kernel(ChainComplexMap) -- kernel of a chain complex map</title>
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<div><h1>kernel(ChainComplexMap) -- kernel of a chain complex 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>kernel f</tt></div>
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<li><span>Function: <a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</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>, a map of chain complexes <tt>C --> D</tt></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 kernel of f</span></li>
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</li>
<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_kernel_lp..._cm_sp__Subring__Limit_sp_eq_gt_sp..._rp.html">SubringLimit => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>If f is <span>a <a href="___Graded__Module__Map.html">graded module map</a></span>, then the result will be <span>a <a href="___Graded__Module.html">graded module</a></span>.<p/>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..d]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I = ideal(a^3,b^3,c^3)
3 3 3
o2 = ideal (a , b , c )
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : C = res coker gens I
1 3 3 1
o3 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o3 : ChainComplex</pre>
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<tr><td><pre>i4 : D = res coker gens (I + ideal(a*b*c))
1 4 6 3
o4 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o4 : ChainComplex</pre>
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<tr><td><pre>i5 : F = extend(D,C,map(D_0,C_0,1))
1 1
o5 = 0 : R <--------- R : 0
| 1 |
4 3
1 : R <----------------- R : 1
{3} | 1 0 0 |
{3} | 0 1 0 |
{3} | 0 0 0 |
{3} | 0 0 1 |
6 3
2 : R <----------------- R : 2
{5} | 0 0 0 |
{5} | 0 0 0 |
{5} | 0 0 0 |
{6} | 1 0 0 |
{6} | 0 1 0 |
{6} | 0 0 1 |
3 1
3 : R <--------------- R : 3
{7} | c2 |
{7} | -b2 |
{7} | a2 |
4 : 0 <----- 0 : 4
0
o5 : ChainComplexMap</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_syz.html" title="the syzygy matrix">syz</a> -- the syzygy matrix</span></li>
<li><span><a href="_generic__Skew__Matrix.html" title="make a generic skew symmetric matrix of variables">genericSkewMatrix</a> -- make a generic skew symmetric matrix of variables</span></li>
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