localizeCharacteristicCycle -- the characteristic cycle of the localized $D$-module

Synopsis

• Usage:

  localizeCharacteristicCycle(cc,f), localizeCharacteristicCycle(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 = localizeCharacteristicCycle(cc,x_1)   -- cc of R_{x_1}

o3 = {ideal () => 1, ideal x  => 1}
                            1

o3 : List

i4 : cc12 = localizeCharacteristicCycle(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

See also

• pruneCechComplexCC -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules
• populateCechComplexCC -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules