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

## 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 .