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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_chain_spcomplexes.html" title="">chain complexes</a> > <a href="_extracting_spinformation_spfrom_spchain_spcomplexes.html" title="">extracting information from chain complexes</a></div>
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<div><h1>extracting information from chain complexes</h1>
<div>Let's make a chain complex.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y,z];</pre>
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<tr><td><pre>i2 : C = res coker matrix {{x,y^2,z^3}};</pre>
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Some simple functions for discovering the shape of <tt>C</tt>.<ul><li><span><a href="_length_lp__Chain__Complex_rp.html" title="length of a chain complex or graded module">length(ChainComplex)</a> -- length of a chain complex or graded module</span></li>
<li><span><a href="_max_lp__Graded__Module_rp.html" title="maximum of elements of a list">max(GradedModule)</a> -- maximum of elements of a list</span></li>
<li><span><a href="_min_lp__Graded__Module_rp.html" title="minimum of elements of a list">min(GradedModule)</a> -- minimum of elements of a list</span></li>
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<table class="examples"><tr><td><pre>i3 : length C
o3 = 3</pre>
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<tr><td><pre>i4 : max C
o4 = 4</pre>
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<tr><td><pre>i5 : min C
o5 = 0</pre>
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In order to see the matrices of the differential maps in a chain complex, examine <tt>C.dd</tt>.<table class="examples"><tr><td><pre>i6 : C.dd
1 3
o6 = 0 : R <--------------- R : 1
| x y2 z3 |
3 3
1 : R <----------------------- R : 2
{1} | -y2 -z3 0 |
{2} | x 0 -z3 |
{3} | 0 x y2 |
3 1
2 : R <--------------- R : 3
{3} | z3 |
{4} | -y2 |
{5} | x |
1
3 : R <----- 0 : 4
0
o6 : ChainComplexMap</pre>
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If <tt>C</tt> is a chain complex, then <tt>C_i</tt> will produce the <tt>i</tt>-th module in the complex, <tt>C^i</tt> will produce the <tt>-i</tt>-th module in it, and <tt>C.dd_i</tt> will produce the differential whose source is <tt>C_i</tt>.<table class="examples"><tr><td><pre>i7 : C_1
3
o7 = R
o7 : R-module, free, degrees {1, 2, 3}</pre>
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<tr><td><pre>i8 : C^-1
3
o8 = R
o8 : R-module, free, degrees {1, 2, 3}</pre>
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<tr><td><pre>i9 : C.dd_2
o9 = {1} | -y2 -z3 0 |
{2} | x 0 -z3 |
{3} | 0 x y2 |
3 3
o9 : Matrix R <--- R</pre>
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The function <a href="_betti.html" title="display degrees">betti</a> can be used to display the ranks of the free modules in <tt>C</tt>, together with the distribution of the basis elements by degree, at least for resolutions of homogeneous modules.<table class="examples"><tr><td><pre>i10 : betti C
0 1 2 3
o10 = total: 1 3 3 1
0: 1 1 . .
1: . 1 1 .
2: . 1 1 .
3: . . 1 1
o10 : BettiTally</pre>
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The ranks are displayed in the top row, and below that in row <tt>i</tt> column <tt>j</tt> is displayed the number of basis elements of degree <tt>i+j</tt> in <tt>C_j</tt>.</div>
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