# BMM -- the characteristic cycle of the localized $D$-module

## Synopsis

• Usage:
BMM(cc,f), BMM(I,cc)
• Inputs:
• cc, a list, the characteristic cycle of a regular holonomic D-module $M$
• I, an ideal, representing an simple' cc
• Outputs:
• List, the characteristic cycle of the localized module $M_f = M[f^{-1}]$

## Description

Provided a characteristic cycle in the form {I_1 => m_1, ..., I_k => m_k} with associated prime ideals I_1,...,I_k and the multiplicities m_1,...,m_k of M along them, the routine computes the characteristic cycle of M_f.

The method is based on a geometric formula given by V.Ginsburg in Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327--402. and reinterpreted by J.Briancon, P.Maisonobe and M.Merle in Localisation de systemes differentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531--550.

 i1 : A = QQ[x_1,x_2,a_1,a_2] o1 = A o1 : PolynomialRing i2 : cc = {ideal A => 1} -- the characteristic ideal of R = CC[x_1,x_2] o2 = {ideal () => 1} o2 : List i3 : cc1 = BMM(cc,x_1) -- cc of R_{x_1} o3 = {ideal () => 1, ideal x => 1} 1 o3 : List 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`

## Caveat

The module has to be a regular holonomic complex-analytic module; while the holomicity can be checked by isHolonomic there is no algorithm to check the regularity.