<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>DHom -- D-homomorphisms between holonomic D-modules</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___D__Hom_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">next</a> | <a href="___D__Ext_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">previous</a> | <a href="___D__Hom_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">forward</a> | <a href="___D__Ext_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>DHom -- D-homomorphisms between holonomic D-modules</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>DHom(M,N), DHom(M,N,w), DHom(I,J)</tt></div>
</dd></dl>
</div>
</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>N</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>J</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which represents the module <em>N = D/J</em></span></li>
<li><span><tt>w</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a positive weight vector</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, a basis of D-homomorphisms between holonomic D-modules <em>M</em> and <em>N</em></span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="___D__Hom_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>The set of D-homomorphisms between two holonomic modules <em>M</em> and <em>N</em> is a finite-dimensional vector space over the ground field. Since a homomorphism is defined by where it sends a set of generators, the output of this command is a list of matrices whose columns correspond to the images of the generators of <em>M</em>. Here the generators of <em>M</em> are determined from its presentation by generators and relations.<p>The procedure calls <a href="___Drestriction.html" title="restriction modules of a D-module">Drestriction</a>, which uses <em>w</em> if specified.</p>
<p>The algorithm used appears in the paper 'Computing homomorphisms between holonomic D-modules' by Tsai-Walther(2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm.</p>
<table class="examples"><tr><td><pre>i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}]
o1 = W
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : M = W^1/ideal(D-1)
o2 = cokernel | D-1 |
1
o2 : W-module, quotient of W</pre>
</td></tr>
<tr><td><pre>i3 : N = W^1/ideal((D-1)^2)
o3 = cokernel | D2-2D+1 |
1
o3 : W-module, quotient of W</pre>
</td></tr>
<tr><td><pre>i4 : DHom(M,N)
o4 = {| -xD+x+1 |, | -D+1 |}
o4 : List</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div>Input modules <em>M</em>, <em>N</em>, <em>D/I</em> and <em>D/J</em> should be holonomic.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="___D__Ext.html" title="Ext groups between holonomic modules">DExt</a> -- Ext groups between holonomic modules</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>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>DHom</tt> :</h2>
<ul><li>DHom(Ideal,Ideal)</li>
<li>DHom(Module,Module)</li>
<li>DHom(Module,Module,List)</li>
</ul>
</div>
</div>
</body>
</html>
