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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_chain_spcomplexes.html" title="">chain complexes</a> > <a href="_maps_spbetween_spchain_spcomplexes.html" title="">maps between chain complexes</a></div>
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<div><h1>maps between chain complexes</h1>
<div>One way to make maps between chain complexes is by lifting maps between modules to resolutions of those modules. First we make some modules.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
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<tr><td><pre>i2 : M = coker vars R
o2 = cokernel | x y |
1
o2 : R-module, quotient of R</pre>
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<tr><td><pre>i3 : N = coker matrix {{x}}
o3 = cokernel | x |
1
o3 : R-module, quotient of R</pre>
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Let's construct the natural map from <tt>N</tt> to <tt>M</tt>.<table class="examples"><tr><td><pre>i4 : f = inducedMap(M,N)
o4 = | 1 |
o4 : Matrix</pre>
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Let's lift the map to a map of free resolutions.<table class="examples"><tr><td><pre>i5 : g = res f
1 1
o5 = 0 : R <--------- R : 0
| 1 |
2 1
1 : R <------------- R : 1
{1} | 1 |
{1} | 0 |
1
2 : R <----- 0 : 2
0
o5 : ChainComplexMap</pre>
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We can check that it's a map of chain complexes this way.<table class="examples"><tr><td><pre>i6 : g * (source g).dd == (target g).dd * g
o6 = true</pre>
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We can form the mapping cone of <tt>g</tt>.<table class="examples"><tr><td><pre>i7 : F = cone g
1 3 2
o7 = R <-- R <-- R <-- 0
0 1 2 3
o7 : ChainComplex</pre>
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Since <tt>f</tt> is surjective, we know that <tt>F</tt> is quasi-isomorphic to <tt>(kernel f)[-1]</tt>. Let's check that.<table class="examples"><tr><td><pre>i8 : prune HH_0 F
o8 = 0
o8 : R-module</pre>
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<tr><td><pre>i9 : prune HH_1 F
o9 = cokernel {1} | x |
1
o9 : R-module, quotient of R</pre>
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<tr><td><pre>i10 : prune kernel f
o10 = cokernel {1} | x |
1
o10 : R-module, quotient of R</pre>
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There are more elementary ways to make maps between chain complexes. The identity map is available from <a href="_id.html" title="identity map">id</a>.<table class="examples"><tr><td><pre>i11 : C = res M
1 2 1
o11 = R <-- R <-- R <-- 0
0 1 2 3
o11 : ChainComplex</pre>
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<tr><td><pre>i12 : id_C
1 1
o12 = 0 : R <--------- R : 0
| 1 |
2 2
1 : R <--------------- R : 1
{1} | 1 0 |
{1} | 0 1 |
1 1
2 : R <------------- R : 2
{2} | 1 |
3 : 0 <----- 0 : 3
0
o12 : ChainComplexMap</pre>
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<tr><td><pre>i13 : x * id_C
1 1
o13 = 0 : R <--------- R : 0
| x |
2 2
1 : R <--------------- R : 1
{1} | x 0 |
{1} | 0 x |
1 1
2 : R <------------- R : 2
{2} | x |
3 : 0 <----- 0 : 3
0
o13 : ChainComplexMap</pre>
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We can use <a href="_induced__Map.html" title="compute an induced map">inducedMap</a> or <tt>**</tt> to construct natural maps between chain complexes.<table class="examples"><tr><td><pre>i14 : inducedMap(C ** R^1/x,C)
1
o14 = 0 : cokernel | x | <--------- R : 0
| 1 |
2
1 : cokernel {1} | x 0 | <--------------- R : 1
{1} | 0 x | {1} | 1 0 |
{1} | 0 1 |
1
2 : cokernel {2} | x | <------------- R : 2
{2} | 1 |
o14 : ChainComplexMap</pre>
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<tr><td><pre>i15 : g ** R^1/x
o15 = 0 : cokernel | x | <--------- cokernel | x | : 0
| 1 |
1 : cokernel {1} | x 0 | <------------- cokernel {1} | x | : 1
{1} | 0 x | {1} | 1 |
{1} | 0 |
o15 : ChainComplexMap</pre>
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There is a way to make a chain complex map by calling a function for each spot that needs a map.<table class="examples"><tr><td><pre>i16 : q = map(C,C,i -> (i+1) * id_(C_i))
1 1
o16 = 0 : R <--------- R : 0
| 1 |
2 2
1 : R <--------------- R : 1
{1} | 2 0 |
{1} | 0 2 |
1 1
2 : R <------------- R : 2
{2} | 3 |
3 : 0 <----- 0 : 3
0
o16 : ChainComplexMap</pre>
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Of course, the formula we used doesn't yield a map of chain complexes.<table class="examples"><tr><td><pre>i17 : C.dd * q == q * C.dd
o17 = false</pre>
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