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intersectionCohom.m2
doc /// Node Key intersectionCohomology IH (intersectionCohomology, Ideal) (intersectionCohomology, ZZ, Ideal) [intersectionCohomology, Strategy] [intersectionCohomology, LocStrategy] [intersectionCohomology, LocCohomStrategy] Headline intersection cohomology of an irreducible affine variety Usage IH I intersectionCohomology(I) IH^d I intersectionCohomology(d,I) Inputs I:Ideal in the polynomial ring $R$ d:ZZ the degree Strategy=>String see @TO [Dintegration, Strategy]@ LocCohomStrategy=>Sequence -- (String, String) see @TO [localCohom, Strategy]@ and @TO [localCohom, LocStrategy]@ LocStrategy=>String see @TO [IHmodule, LocStrategy]@ Outputs :HashTable a table including the intersection cohomology groups of the irreducible variety Spec($R/I$), :Module the intersection cohomology group in degree d Description Text This routine computes the middle intersection cohomology groups of the irreducible variety defined by $I$ in the affine space Spec($R$). Example R=QQ[x,y] I = ideal(x^2+y^3) intersectionCohomology(I) Caveat Must be over a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime. Node Key IHmodule (IHmodule, Ideal) [IHmodule, LocStrategy] [IHmodule, LocCohomStrategy] LocCohomStrategy Headline intersection (co)homology module of an irreducible closed subvariety Usage IHmodule(I) Inputs I:Ideal ideal in the polynomial ring $R$ LocCohomStrategy=>Sequence -- (String, String) see @TO [localCohom, Strategy]@ and @TO[localCohom, LocStrategy]@ LocStrategy=>String see @TO [Dlocalize, Strategy]@, or for regular sequence use CompleteIntersection Outputs :Module the intersection cohomology $D$-module Description Text 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. Example R=QQ[x,y,z] I=ideal(x^2+y^3) IHmodule(I) Text 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. Example R=QQ[x,y] I=ideal(x^2+y^3) IHmodule(I, LocStrategy=>CompleteIntersection) Caveat Must be a ring of characteristic 0. The ideal $I$ should have only 1 minimal prime. ///
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