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<div><h1>topComponents -- compute top dimensional component</h1>
<div class="single"><h2>Description</h2>
<div>The method used is that of Eisenbud-Huneke-Vasconcelos, in their 1993 Inventiones Mathematicae paper.<p/>
If M is a module in a polynomial ring R, then the implementations of <a href="_top__Components.html" title="compute top dimensional component">topComponents</a> and <a href="_remove__Lowest__Dimension.html" title="remove components of lowest dimension">removeLowestDimension</a> are based on the following observations:<ul><li><i>codim Ext<sup>d</sup>(M,R) ≥d</i> for all d</li>
<li>If <i>P</i> is an associated prime of <i>M</i> of codimension <i>d := codim P > codim M</i>, then <i>codim Ext<sup>d</sup>(M,R) = d</i> and the annihilator of <i>Ext<sup>d</sup>(M,R)</i> is contained in <i>P</i></li>
<li>If <i>codim Ext<sup>d</sup>(M,R) = d</i>, then there really is an associated prime of codimension <i>d</i>.</li>
<li>If <i>M</i> is <i>R/I</i>, then <i>topComponents(I) = ann Ext<sup>c</sup>(R/I,R)</i>, where <i>c = codim I</i></li>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_remove__Lowest__Dimension.html" title="remove components of lowest dimension">removeLowestDimension</a> -- remove components of lowest dimension</span></li>
<li><span><a href="_saturate.html" title="saturation of ideal or submodule">saturate</a> -- saturation of ideal or submodule</span></li>
<li><span><a href="_quotient.html" title="quotient or division">quotient</a> -- quotient or division</span></li>
<li><span><a href="_radical.html" title="the radical of an ideal">radical</a> -- the radical of an ideal</span></li>
<li><span><a href="_component_spexample.html" title="">component example</a></span></li>
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<div class="waystouse"><h2>Ways to use <tt>topComponents</tt> :</h2>
<ul><li><span><a href="_top__Components_lp__Ideal_rp.html" title="compute top dimensional component">topComponents(Ideal)</a> -- compute top dimensional component</span></li>
<li><span><a href="_top__Components_lp__Module_rp.html" title="compute top dimensional component">topComponents(Module)</a> -- compute top dimensional component</span></li>
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