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<head><title>HH^ZZ ChainComplex -- cohomology of a chain complex</title>
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<div><h1>HH^ZZ ChainComplex -- cohomology of a chain complex</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>HH^i C</tt></div>
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<li><span>Function: <a href="_cohomology.html" title="general cohomology functor">cohomology</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>C</tt>, <span>a <a href="___Chain__Complex.html">chain complex</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, HH^i C -- homology at the i-th spot of the chain complex <tt>C</tt>.</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_cohomology_lp..._cm_sp__Degree_sp_eq_gt_sp..._rp.html">Degree => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>By definition, this is the same as computing HH_(-i) C.<p/>
<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : C = chainComplex(matrix{{x,y}},matrix{{x*y},{-x^2}})
1 2 1
o2 = R <-- R <-- R
0 1 2
o2 : ChainComplex</pre>
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<tr><td><pre>i3 : M = HH^1 C
o3 = 0
o3 : R-module</pre>
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<tr><td><pre>i4 : prune M
o4 = 0
o4 : R-module</pre>
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Here is another example computing simplicial cohomology (for a hollow tetrahedron):<table class="examples"><tr><td><pre>i5 : needsPackage "SimplicialComplexes"
o5 = SimplicialComplexes
o5 : Package</pre>
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<tr><td><pre>i6 : R = QQ[a..d]
o6 = R
o6 : PolynomialRing</pre>
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<tr><td><pre>i7 : D = simplicialComplex {a*b*c,a*b*d,a*c*d,b*c*d}
o7 = | bcd acd abd abc |
o7 : SimplicialComplex</pre>
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<tr><td><pre>i8 : C = chainComplex D
1 4 6 4
o8 = QQ <-- QQ <-- QQ <-- QQ
-1 0 1 2
o8 : ChainComplex</pre>
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<tr><td><pre>i9 : HH_2 C
o9 = image | -1 |
| 1 |
| -1 |
| 1 |
4
o9 : QQ-module, submodule of QQ</pre>
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<tr><td><pre>i10 : prune oo
1
o10 = QQ
o10 : QQ-module, free</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Graded__Module.html" title="the class of all graded modules">GradedModule</a> -- the class of all graded modules</span></li>
<li><span><a href="___H__H.html" title="general homology and cohomology functor">HH</a> -- general homology and cohomology functor</span></li>
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