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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_chain_spcomplexes.html" title="">chain complexes</a> > <a href="_making_spchain_spcomplexes_spby_sphand.html" title="">making chain complexes by hand</a></div>
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<div><h1>making chain complexes by hand</h1>
<div>A new chain complex can be made with <tt>C = new ChainComplex</tt>. This will automatically initialize <tt>C.dd</tt>, in which the differentials are stored. The modules can be installed with statements like <tt>C#i=M</tt> and the differentials can be installed with statements like <tt>C.dd#i=d</tt>. The ring is installed with <tt>C.ring = R</tt>. It's up to the user to ensure that the composite of consecutive differential maps is zero.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : d1 = matrix {{x,y}};
1 2
o2 : Matrix R <--- R</pre>
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We take care to use <a href="_map.html" title="make a map">map</a> to ensure that the target of <tt>d2</tt> is exactly the same as the source of <tt>d1</tt>.<table class="examples"><tr><td><pre>i3 : d2 = map(source d1, ,{{y*z},{-x*z}});
2 1
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : d1 * d2 == 0
o4 = true</pre>
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Now we make the chain complex, as explained above.<table class="examples"><tr><td><pre>i5 : C = new ChainComplex; C.ring = R;</pre>
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<tr><td><pre>i7 : C#0 = target d1; C#1 = source d1; C#2 = source d2;</pre>
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<tr><td><pre>i10 : C.dd#1 = d1; C.dd#2 = d2;
1 2
o10 : Matrix R <--- R
2 1
o11 : Matrix R <--- R</pre>
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Our complex is ready to use.<table class="examples"><tr><td><pre>i12 : C
1 2 1
o12 = R <-- R <-- R
0 1 2
o12 : ChainComplex</pre>
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<tr><td><pre>i13 : HH_0 C
o13 = cokernel | x y |
1
o13 : R-module, quotient of R</pre>
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<tr><td><pre>i14 : prune HH_1 C
o14 = cokernel {2} | z |
1
o14 : R-module, quotient of R</pre>
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The chain complex we've just made is simple, in the sense that it's a homological chain complex with nonzero modules in degrees 0, 1, ..., n. Such a chain complex can be made also with <a href="_chain__Complex.html" title="make a chain complex">chainComplex</a>. It goes to a bit of extra trouble to adjust the differentials to match the degrees of the basis elements.<table class="examples"><tr><td><pre>i15 : D = chainComplex(matrix{{x,y}}, matrix {{y*z},{-x*z}})
1 2 1
o15 = R <-- R <-- R
0 1 2
o15 : ChainComplex</pre>
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<tr><td><pre>i16 : degrees source D.dd_2
o16 = {{3}}
o16 : List</pre>
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