IHmodule -- intersection (co)homology module of an irreducible closed subvariety

Synopsis

Usage:

IHmodule(I)
Inputs:
- I, an ideal, ideal in the polynomial ring $R$
Optional inputs:
- LocCohomStrategy => a sequence, default value (Walther,), (String, String); see localCohom(...,Strategy=>...) and localCohom(...,LocStrategy=>...)
- LocStrategy => a string, default value OTW, see Dlocalize(...,Strategy=>...), or for regular sequence use CompleteIntersection
Outputs:
- a module, the intersection cohomology $D$-module

Description

This routine gives a presentation of the Brylinski-Kashiwara intersection cohomology $D$-module of the closed subvariety defined by $I$. Via the Riemann-Hilbert correspondence, this corresponds to the trivial local system on the smooth locus of the variety.

i1 : R=QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : I=ideal(x^2+y^3)

            3    2
o2 = ideal(y  + x )

o2 : Ideal of R

i3 : IHmodule(I)

o3 = subquotient (| 0  0  |, | dz -x 0  y2 -2ydy-6 0            |)
                  | xy x2 |  | 0  y  dz x  -3ydx   -3xdx-2ydy-8 |

                                                                       2
o3 : QQ[x..z, dx, dy, dz]-module, subquotient of (QQ[x..z, dx, dy, dz])

When the given generators of $I$ form a regular sequence, use LocStrategy=>CompleteIntersection for a generally faster algorithm, which implements the determination of the IC module in terms of the fundamental class as described in: D. Barlet and M. Kashiwara, Le réseau $L^2$ d’un système holonome régulier, Invent. Math. 86 (1986), no. 1, 35–62.

i4 : R=QQ[x,y]

o4 = R

o4 : PolynomialRing

i5 : I=ideal(x^2+y^3)

            3    2
o5 = ideal(y  + x )

o5 : Ideal of R

i6 : IHmodule(I, LocStrategy=>CompleteIntersection)

o6 = subquotient (| x |, | 3xdx+2ydy+6 3y2dx-2xdy y3+x2 |)

                                                               1
o6 : QQ[x..y, dx, dy]-module, subquotient of (QQ[x..y, dx, dy])

Caveat

Must be a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime.

Ways to use IHmodule :

IHmodule(Ideal)

For the programmer

The object IHmodule is a method function with options.