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Dmodules :: charIdeal

charIdeal -- characteristic ideal of a D-module

Synopsis

Description

The characteristic ideal of M is the annihilator of gr(M) under a good filtration with respect to the order filtration. If D is the Weyl algebra over ℂ with generators x1,…,xn and 1,…,∂n, then the order filtration corresponds to the weight vector (0,...,0,1,...,1). The characteristic ideal lives in the associated graded ring of D with respect to the order filtration, and this is a commutative polynomial ring ℂ[x1,…,xn1,…,ξn]. Here the ξi is the principal symbol of i, that is, the image of i in the associated graded ring. The zero locus of the characteristic ideal is equal to the characteristic variety of D/I which is an invariant of a D-module.

The algorithm to compute the characteristic ideal consists of computing the initial ideal of I with respect to the weight vector (0,...,0,1...,1). More details can be found in [SST, Section 1.4].

i1 : makeWA(QQ[x,y])

o1 = QQ[x, y, dx, dy]

o1 : PolynomialRing, 2 differential variables
i2 : I = ideal (x*dx+2*y*dy-3, dx^2-dy)

                                2
o2 = ideal (x*dx + 2y*dy - 3, dx  - dy)

o2 : Ideal of QQ[x, y, dx, dy]
i3 : charIdeal I

              2
o3 = ideal (dx , x*dx + 2y*dy)

o3 : Ideal of QQ[x, y, dx, dy]

See also

Ways to use charIdeal :