<?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>PolyExt -- Ext groups between a holonomic module and a polynomial ring</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Poly__Ext_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">next</a> | <a href="___Poly__Ann_lp__Ring__Element_rp.html">previous</a> | <a href="___Poly__Ext_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">forward</a> | <a href="___Poly__Ann_lp__Ring__Element_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>PolyExt -- Ext groups between a holonomic module and a polynomial ring</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>PolyExt M, PolyExt I; RatExt(i,M), RatExt(i,I)</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>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>i</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, nonnegative</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> or <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, the Ext<sup>i</sup> group(s) between holonomic <em>M</em> and the polynomial ring</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="___Poly__Ext_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 <tt>Ext</tt> groups between a D-module <em>M</em> and the polynomial ring are the derived functors of <tt>Hom</tt>, and are finite-dimensional vector spaces over the ground field when <em>M</em> is holonomic.<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>i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}]
o1 = W
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : M = W^1/ideal(x^2*D^2)
o2 = cokernel | x2D2 |
1
o2 : W-module, quotient of W</pre>
</td></tr>
<tr><td><pre>i3 : PolyExt(M)
2
o3 = HashTable{0 => QQ }
2
1 => QQ
o3 : HashTable</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div>Does not yet compute explicit representations of Ext groups such as Yoneda representation.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Poly__Sols.html" title="polynomial solutions of a holonomic system">PolySols</a> -- polynomial solutions of a holonomic system</span></li>
<li><span><a href="___Rat__Ext.html" title="Ext(holonomic D-module, polynomial ring localized at the sigular locus)">RatExt</a> -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)</span></li>
<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="___Dintegration.html" title="integration modules of a D-module">Dintegration</a> -- integration modules of a D-module</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>PolyExt</tt> :</h2>
<ul><li>PolyExt(Ideal)</li>
<li>PolyExt(Module)</li>
<li>PolyExt(ZZ,Ideal)</li>
<li>PolyExt(ZZ,Module)</li>
</ul>
</div>
</div>
</body>
</html>
