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<head><title>DExt -- Ext groups between holonomic modules</title>
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<div><h1>DExt -- Ext groups between holonomic 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>DExt(M,N), DExt(M,N,w)</tt></div>
</dd></dl>
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<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>w</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a positive weight vector</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, the <tt>Ext</tt> groups between holonomic D-modules<em>M</em> and <em>N</em></span></li>
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</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="___D__Ext_lp..._cm_sp__Info_sp_eq_gt_sp..._rp.html">Info => ...</a>, </span></li>
<li><span><a href="___D__Ext_lp..._cm_sp__Output_sp_eq_gt_sp..._rp.html">Output => ...</a>, </span></li>
<li><span><a href="___D__Ext_lp..._cm_sp__Special_sp_eq_gt_sp..._rp.html">Special => ...</a>, </span></li>
<li><span><a href="___D__Ext_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>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 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 '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*(D-1))
o2 = cokernel | xD-x |
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>
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<tr><td><pre>i4 : DExt(M,N)
2
o4 = HashTable{0 => QQ }
2
1 => QQ
o4 : HashTable</pre>
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<div class="single"><h2>Caveat</h2>
<div>Input modules M, N should be holonomic.Does not yet compute explicit reprentations of Ext groups such as Yoneda representation.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___D__Hom.html" title="D-homomorphisms between holonomic D-modules">DHom</a> -- D-homomorphisms between holonomic D-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>
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<div class="waystouse"><h2>Ways to use <tt>DExt</tt> :</h2>
<ul><li>DExt(Module,Module)</li>
<li>DExt(Module,Module,List)</li>
</ul>
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