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<head><title>computing resolutions</title>
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<div><h1>computing resolutions</h1>
<div>Use the function <a href="_resolution.html" title="projective resolution">resolution</a>, often abbreviated as <tt>res</tt>, to compute a free resolution of a module.<table class="examples"><tr><td><pre>i1 : R = QQ[x..z];</pre>
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<tr><td><pre>i2 : M = cokernel vars R
o2 = cokernel | x y z |
1
o2 : R-module, quotient of R</pre>
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<tr><td><pre>i3 : C = res M
1 3 3 1
o3 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o3 : ChainComplex</pre>
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See <a href="_chain_spcomplexes.html" title="">chain complexes</a> for further details about how to handle and examine the result.<p/>
A reference to the result is stored within the module <tt>M</tt>, so that requesting a computation of <tt>res M</tt> a second time yields the formerly computed result immediately.<p/>
If the computation is interrupted or discontinued after the skeleton has been successfully computed, then the partially completed resolution is available as <tt>M.cache.resolution</tt>, and can be examined with <a href="_status.html" title="status of a resolution computation">status</a>. The computation can be continued with <tt>res M</tt>. Here is an example, with an alarm interrupting the computation several times before it's complete. (On my machine, the computation takes a total of 14 seconds.)<p/>
<table class="examples"><tr><td><pre>i4 : R = ZZ/2[a..d];</pre>
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<tr><td><pre>i5 : M = coker random(R^4, R^{5:-3,6:-4});</pre>
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<tr><td><pre>i6 : (<< "-- computation started: " << endl;
while true do try (
alarm 3;
time res M;
alarm 0;
<< "-- computation complete" << endl;
status M.cache.resolution;
<< res M << endl << endl;
break;
) else (
<< "-- computation interrupted" << endl;
status M.cache.resolution;
<< "-- continuing the computation" << endl;
))
-- computation started:
-- used 1.3528 seconds
-- computation complete
4 11 89 122 40
R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5</pre>
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If the user has a chain complex in hand that is known to be a projective resolution of <tt>M</tt>, then it can be installed with <tt>M.cache.resolution = C</tt>.<p/>
There are various optional arguments associated with <a href="_resolution.html" title="projective resolution">resolution</a> which allow detailed control over the progress of the computation.</div>
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