charIdeal -- characteristic ideal of a D-module

Synopsis

• Usage:
charIdeal M, charIdeal I
• Inputs:
• M, , over the Weyl algebra D
• I, an ideal, which represents the module M = D/I
• Outputs:
• an ideal, the characteristic ideal of M

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 &#x2102; with generators $x_1,\dots,x_n$ and $\partial_1,\dots,\partial_n$, 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 &#x2102;$[x_1,\dots,x_n,\xi_1,\dots,\xi_n]$. Here the $\xi_i$ is the principal symbol of $\partial_i$, that is, the image of $\partial_i$ 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]