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<head><title>BMM -- the characteristic cycle of the localized $D$-module</title>
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<div><h1>BMM -- the characteristic cycle of the localized $D$-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>BMM(cc,f), BMM(I,cc)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>cc</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, the characteristic cycle of a regular holonomic D-module <i>M</i></span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, representing an `simple' <tt>cc</tt></span></li>
</ul>
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<li><div class="single">Outputs:<ul><li><span><tt>List</tt>, the characteristic cycle of the localized module <i>M<sub>f</sub> = M[f<sup>-1</sup>]</i></span></li>
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<div class="single"><h2>Description</h2>
<div><p>Provided a characteristic cycle in the form <tt>{I_1 => m_1, ..., I_k => m_k}</tt> with associated prime ideals I<sub>1</sub>,...,I<sub>k</sub> and the multiplicities m<sub>1</sub>,...,m<sub>k</sub> of M along them, the routine computes the characteristic cycle of M<sub>f</sub>.</p>
<p>The method is based on a geometric formula given by V.Ginsburg in <em>Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327--402.</em> and reinterpreted by J.Briancon, P.Maisonobe and M.Merle in <em>Localisation de systemes differentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531--550</em>.</p>
<table class="examples"><tr><td><pre>i1 : A = QQ[x_1,x_2,a_1,a_2]
o1 = A
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : cc = {ideal A => 1} -- the characteristic ideal of R = CC[x_1,x_2]
o2 = {ideal () => 1}
o2 : List</pre>
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<tr><td><pre>i3 : cc1 = BMM(cc,x_1) -- cc of R_{x_1}
o3 = {ideal () => 1, ideal(x ) => 1}
1
o3 : List</pre>
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<tr><td><pre>i4 : cc12 = BMM(cc1,x_2) -- cc of R_{x_1x_2}
o4 = {ideal () => 1, ideal(x ) => 1, ideal(x ) => 1, ideal (x , x ) => 1}
2 1 2 1
o4 : List</pre>
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<div class="single"><h2>Caveat</h2>
<div>The module has to be a regular holonomic complex-analytic module; while the holomicity can be checked by <a href="_is__Holonomic.html" title="determines whether a D-module (or ideal in Weyl algebra) is holonomic">isHolonomic</a> there is no algorithm to check the regularity.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_prune__Cech__Complex__C__C_lp__Mutable__Hash__Table_rp.html" title="reduction of the Cech complex that produces characteristic cycles of local cohomology modules">pruneCechComplexCC</a> -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules</span></li>
<li><span><a href="_populate__Cech__Complex__C__C_lp__Ideal_cm__List_rp.html" title="Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules">populateCechComplexCC</a> -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>BMM</tt> :</h2>
<ul><li>BMM(Ideal,RingElement)</li>
<li>BMM(List,RingElement)</li>
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