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<!DOCTYPE html> <html lang="en"> <head> <title>DExt -- Ext groups between holonomic modules</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="Ext groups between holonomic modules" href="___D__Ext.html">DExt</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="___D__Ext_lp..._cm__Info_eq_gt..._rp.html">next</a> | <a href="_de__Rham__All_lp__Ring__Element_rp.html">previous</a> | <a href="___D__Ext_lp..._cm__Info_eq_gt..._rp.html">forward</a> | <a href="_de__Rham__All_lp__Ring__Element_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>DExt -- Ext groups between holonomic modules</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">DExt(M,N), DExt(M,N,w)</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">N</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">w</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, a positive weight vector</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a href="___D__Ext_lp..._cm__Info_eq_gt..._rp.html">Info</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value 1</span>, <span></span></span></li> <li><span><a href="___D__Ext_lp..._cm__Output_eq_gt..._rp.html">Output</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value HomologyModules</span>, <span></span></span></li> <li><span><a href="___D__Ext_lp..._cm__Special_eq_gt..._rp.html">Special</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value None</span>, <span></span></span></li> <li><span><a href="___D__Ext_lp..._cm__Strategy_eq_gt..._rp.html">Strategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value Schreyer</span>, <span></span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, the <span class="tt">Ext</span> groups between holonomic D-modules<em>M</em> and <em>N</em></span></li> </ul> </li> </ul> <div> <h2>Description</h2> The Ext groups between D-modules <em>M</em> and <em>N</em> are the derived functors of Hom, and are finite-dimensional vector spaces over the ground field when <em>M</em> and <em>N</em> are holonomic. <p>The procedure calls <a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a>, which uses <em>w</em> if specified.</p> <p>The algorithm used appears in the paper 'Polynomial and rational solutions of holonomic systems' by Oaku-Takayama-Tsai (2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}] o1 = W o1 : PolynomialRing, 1 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : M = W^1/ideal(x*(D-1)) o2 = cokernel | xD-x | 1 o2 : W-module, quotient of W</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : N = W^1/ideal((D-1)^2) o3 = cokernel | D2-2D+1 | 1 o3 : W-module, quotient of W</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : DExt(M,N) 2 o4 = HashTable{0 => QQ } 2 1 => QQ o4 : HashTable</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> Input modules M, N should be holonomic.Does not yet compute explicit representations of Ext groups such as Yoneda representation. </div> <div> <h2>See also</h2> <ul> <li><span><a title="D-homomorphisms between holonomic D-modules" href="___D__Hom.html">DHom</a> -- D-homomorphisms between holonomic D-modules</span></li> <li><span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">DExt</span>:</h2> <ul> <li><kbd>DExt(Module,Module)</kbd></li> <li><kbd>DExt(Module,Module,List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Ext groups between holonomic modules" href="___D__Ext.html">DExt</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/DHom.m2:116:0</span>.</p> </div> </div> </div> </body> </html>