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<head><title>Dintegration -- integration modules of a D-module</title>
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<div><h1>Dintegration -- integration modules of a D-module</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>N = Dintegration(M,w), NI = Dintegration(I,w), Ni = Dintegration(i,M,w),</tt><br/><tt> NIi = Dintegration(i,I,w), </tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, over the Weyl algebra <em>D</em></span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which represents the module <em>M = D/I</em></span></li>
<li><span><tt>w</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a weight vector</span></li>
<li><span><tt>i</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, nonnegative</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>Ni</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, the i-th derived integration module of <em>M</em> with respect to the weight vector <em>w</em></span></li>
<li><span><tt>N</tt>, <span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, contains entries of the form <tt>i=>Ni</tt></span></li>
<li><span><tt>NIi</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, the i-th derived integration module of <em>D/I</em> with respect to the weight vector <em>w</em></span></li>
<li><span><tt>NI</tt>, <span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, contains entries of the form <tt>i=>NIi</tt></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="___Dintegration_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>The derived integration modules of a D-module <em>M</em> are the derived direct images in the category of D-modules. This routine computes integration for projection to coordinate subspaces, where the subspace is determined by the strictly positive entries of the weight vector <em>w</em>, e.g., <em>{x_i = 0 : w_i > 0}</em> if <em>D = </em><b>C</b><em><x_1,...,x_n,d_1,...,d_n></em>. The input weight vector should be a list of <em>n</em> numbers to induce the weight <em>(-w,w)</em> on <em>D</em>.<p></p>
The algorithm used appears in the paper 'Algorithims for D-modules' by Oaku-Takayama(1999). The method is to take the Fourier transform of M, then compute the derived restriction, then inverse Fourier transform back.<table class="examples"><tr><td><pre>i1 : R = QQ[x_1,x_2,D_1,D_2,WeylAlgebra=>{x_1=>D_1,x_2=>D_2}]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I = ideal(x_1, D_2-1)
o2 = ideal (x , D - 1)
1 2
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : Dintegration(I,{1,0})
o3 = HashTable{0 => cokernel | D_2-1 |}
1 => 0
o3 : HashTable</pre>
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<div class="single"><h2>Caveat</h2>
<div>The module M should be specializable to the subspace. This is true for holonomic modules.The weight vector w should be a list of n numbers if M is a module over the nth Weyl algebra.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Dintegration__All.html" title="integration modules of a D-module (extended version)">DintegrationAll</a> -- integration modules of a D-module (extended version)</span></li>
<li><span><a href="___Dintegration__Classes.html" title="integration classes of a D-module">DintegrationClasses</a> -- integration classes of a D-module</span></li>
<li><span><a href="___Dintegration__Complex.html" title="derived integration complex of a D-module">DintegrationComplex</a> -- derived integration complex of a D-module</span></li>
<li><span><a href="___Dintegration__Ideal.html" title="integration ideal of a D-module">DintegrationIdeal</a> -- integration ideal of a D-module</span></li>
<li><span><a href="___Drestriction.html" title="restriction modules of a D-module">Drestriction</a> -- restriction modules of a D-module</span></li>
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<div class="waystouse"><h2>Ways to use <tt>Dintegration</tt> :</h2>
<ul><li>Dintegration(Ideal,List)</li>
<li>Dintegration(Module,List)</li>
<li>Dintegration(ZZ,Ideal,List)</li>
<li>Dintegration(ZZ,Module,List)</li>
</ul>
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