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<!DOCTYPE html> <html lang="en"> <head> <title>Ddual -- holonomic dual of a D-module</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="algorithms for b-functions, local cohomology, and intersection cohomology" href="index.html">BernsteinSato</a> :: <a title="holonomic dual of a D-module" href="___Ddual.html">Ddual</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_de__Rham.html">next</a> | <a href="___Cycles.html">previous</a> | <a href="_de__Rham.html">forward</a> | <a href="___Cycles.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Ddual -- holonomic dual of a D-module</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">Ddual M, Ddual I</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">M</span>, <span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, over the Weyl algebra <em>D</em></span></li> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which represents the module <em>M = D/I</em></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all modules" href="../../Macaulay2Doc/html/___Module.html">module</a></span>, the holonomic dual of <em>M</em></span></li> </ul> </li> </ul> <div> <h2>Description</h2> 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><code class="language-macaulay2">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]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">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])</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> The input module <em>M</em> should be holonomic. The user should check this manually with the script <span class="tt">Ddim</span>. </div> <div> <h2>See also</h2> <ul> <li><span><a title="dimension of a D-module" href="../../WeylAlgebras/html/___Ddim.html">Ddim</a> -- dimension of a D-module</span></li> <li><span><a title="standard transposition for Weyl algebra" href="../../WeylAlgebras/html/___Dtransposition.html">Dtransposition</a> -- standard transposition for Weyl algebra</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">Ddual</span>:</h2> <ul> <li><kbd>Ddual(Ideal)</kbd></li> <li><kbd>Ddual(Module)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="holonomic dual of a D-module" href="___Ddual.html">Ddual</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/DHom.m2:144:0</span>.</p> </div> </div> </div> </body> </html>