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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_chain_spcomplexes.html" title="">chain complexes</a> > <a href="_free_spresolutions_spof_spmodules.html" title="">free resolutions of modules</a></div>
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<div><h1>free resolutions of modules</h1>
<div>The function <a href="_resolution.html" title="projective resolution">resolution</a> (also called <tt>res</tt>), can be used to produce a free resolution of a module.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y];</pre>
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<tr><td><pre>i2 : m = ideal vars R
o2 = ideal (x, y)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : M = m/m^3
o3 = subquotient (| x y |, | x3 x2y xy2 y3 |)
1
o3 : R-module, subquotient of R</pre>
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<tr><td><pre>i4 : C = resolution M
2 5 3
o4 = R <-- R <-- R <-- 0
0 1 2 3
o4 : ChainComplex</pre>
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The default display for a chain complex shows the modules and the number of the stage at which they appear. See the documentation of <a href="_resolution.html" title="projective resolution">resolution</a> for details on the options that can be used to control the computation.<p/>
The same function, applied to a map <tt>f</tt>, will produce a map from a free resolution of the source of <tt>f</tt> to a free resolution of the target of <tt>f</tt>.<table class="examples"><tr><td><pre>i5 : h = resolution inducedMap(M, m^2/m^4)
2 3
o5 = 0 : R <----------------- R : 0
{1} | x y 0 |
{1} | 0 0 y |
5 7
1 : R <------------------------- R : 1
{2} | 0 y 0 0 0 0 0 |
{3} | 0 0 x y 0 0 0 |
{3} | 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 x y 0 |
{3} | 0 0 0 0 0 0 y |
3 4
2 : R <------------------- R : 2
{4} | 0 y 0 0 |
{4} | 0 x 0 0 |
{4} | 0 0 0 y |
3 : 0 <----- 0 : 3
0
o5 : ChainComplexMap</pre>
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