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<head><title>regularity -- compute the Castelnuovo-Mumford regularity</title>
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<div><h1>regularity -- compute the Castelnuovo-Mumford regularity</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>regularity C</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>C</tt>, a <a href="___Chain__Complex.html" title="the class of all chain complexes">ChainComplex</a>, an <a href="___Ideal.html" title="the class of all ideals">Ideal</a>, or a <a href="___Module.html" title="the class of all modules">Module</a></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_regularity_lp..._cm_sp__Weights_sp_eq_gt_sp..._rp.html">Weights => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>For a free chain complex C, the regularity r is the smallest number so that each basis element of C_i has degree at most i+r. For a module M, the regularity is the regularity of a free minimal resolution of M.<table class="examples"><tr><td><pre>i1 : R=ZZ/32003[a..d];</pre>
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<tr><td><pre>i2 : I=ideal(a^20,b^20,a*c^19-b*d^19);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : regularity I
o3 = 399</pre>
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The regularity is the label of the last row in the betti diagram of a chain complex.<table class="examples"><tr><td><pre>i4 : J=ideal(a^3,a^2*b,a*b^6,a^2*c);
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : C=resolution J
1 4 4 1
o5 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o5 : ChainComplex</pre>
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<tr><td><pre>i6 : betti C
0 1 2 3
o6 = total: 1 4 4 1
0: 1 . . .
1: . . . .
2: . 3 3 1
3: . . . .
4: . . . .
5: . . . .
6: . 1 1 .
o6 : BettiTally</pre>
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<tr><td><pre>i7 : regularity C
o7 = 6</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_resolution.html" title="projective resolution">resolution</a> -- projective resolution</span></li>
<li><span><a href="_betti.html" title="display degrees">betti</a> -- display degrees</span></li>
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<div class="waystouse"><h2>Ways to use <tt>regularity</tt> :</h2>
<ul><li><span>regularity(BettiTally), see <span><a href="___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> -- the class of all Betti tallies</span></span></li>
<li>regularity(ChainComplex)</li>
<li>regularity(Ideal)</li>
<li>regularity(Module)</li>
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