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<head><title>Ddual -- holonomic dual of a D-module</title>
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<div><h1>Ddual -- holonomic dual 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>Ddual M, Ddual I</tt></div>
</dd></dl>
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</li>
<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>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, the holonomic dual of <em>M</em></span></li>
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<div class="single"><h2>Description</h2>
<div>If M is a holonomic left D-module, then <b>Ext</b><sup>n</sup><sub>D</sub>(<em>M,D</em>) is a holonomic right D-module. The holonomic dual is defined to be the left module associated to <b>Ext</b><sup>n</sup><sub>D</sub>(<em>M,D</em>). The dual is obtained by computing a free resolution of <em>M</em>, dualizing, and applying the standard transposition to the <em>n</em>-th homology.<table class="examples"><tr><td><pre>i1 : I = AppellF1({1,0,-3,2})
3 2 2 2 2 2
o1 = ideal (- x Dx - x y*Dx*Dy + x Dx + x*y*Dx*Dy - 2x Dx + 2x*Dx, -
------------------------------------------------------------------------
2 3 2 2 2 2
x*y Dx*Dy - y Dy + x*y*Dx*Dy + y Dy + 3x*y*Dx + y Dy + 2y*Dy + 3y,
------------------------------------------------------------------------
x*Dx*Dy - y*Dx*Dy + 3Dx)
o1 : Ideal of QQ[x, y, Dx, Dy]</pre>
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<tr><td><pre>i2 : Ddual I
o2 = cokernel | 0 xDy-yDy-4 x2Dx+y2Dy-xDx-yDy+x+4y y2DxDy+y2Dy^2-yDxDy-yDy^2+4xDx+4yDx+5yDy-4Dx+4 0 |
| Dx -yDy-1 0 0 y3Dy^2-y2Dy^2+7y2Dy-2yDy+5y |
2
o2 : QQ[x, y, Dx, Dy]-module, quotient of (QQ[x, y, Dx, Dy])</pre>
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<div class="single"><h2>Caveat</h2>
<div>The input module <em>M</em> should be holonomic. The user should check this manually with the script <tt>Ddim</tt>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Ddim.html" title="dimension of a D-module">Ddim</a> -- dimension of a D-module</span></li>
<li><span><a href="___Dtransposition.html" title="standard transposition for Weyl algebra">Dtransposition</a> -- standard transposition for Weyl algebra</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>Ddual</tt> :</h2>
<ul><li>Ddual(Ideal)</li>
<li>Ddual(Module)</li>
</ul>
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