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<head><title>Dlocalize -- localization of a D-module</title>
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<div><h1>Dlocalize -- localization of a D-module</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>Dlocalize(M,f), Dlocalize(I,f)</tt></div>
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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>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>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a polynomial</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, the localized module M<sub>f</sub> = M[f<sup>-1</sup>] as a D-module</span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="___Dlocalize_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- strategy for computing a localization of a D-module</span></li>
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<div class="single"><h2>Description</h2>
<div>One of the nice things about D-modules is that if a finitely generated D-module is specializable along <em>f</em>, then it's localization with respect to <em>f</em> is also finitely generated. For instance, this is true for all holonomic D-modules.<p/>
There are two different algorithms for localization implemented. The first appears in the paper 'A localization algorithm for D-modules' by Oaku-Takayama-Walther (1999). The second is due to Oaku and appears in the paper 'Algorithmic computation of local cohomology modules and the cohomological dimension of algebraic varieties' by Walther(1999)<table class="examples"><tr><td><pre>i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}]
o1 = W
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M = W^1/(ideal(x*Dx+1, Dy))
o2 = cokernel | xDx+1 Dy |
1
o2 : W-module, quotient of W</pre>
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<tr><td><pre>i3 : f = x^2-y^3
3 2
o3 = - y + x
o3 : W</pre>
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<tr><td><pre>i4 : Mf = Dlocalize(M, f)
o4 = cokernel | 3xDx+2yDy+15 y3Dy-x2Dy+6y2 |
1
o4 : W-module, quotient of W</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Dlocalize__All.html" title="localization of a D-module (extended version)">DlocalizeAll</a> -- localization of a D-module (extended version)</span></li>
<li><span><a href="___Dlocalize__Map.html" title="localization map from a D-module to its localization">DlocalizeMap</a> -- localization map from a D-module to its localization</span></li>
<li><span><a href="___Ann__Fs.html" title="the annihilating ideal of f^s">AnnFs</a> -- the annihilating ideal of f^s</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>
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<div class="waystouse"><h2>Ways to use <tt>Dlocalize</tt> :</h2>
<ul><li>Dlocalize(Ideal,RingElement)</li>
<li>Dlocalize(Module,RingElement)</li>
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