7.3.3 dim (plural)

Syntax:

dim ( ideal_expression )
dim ( module_expression )

Type:

int

Purpose:

computes the Gelfand-Kirillov dimension of the ideal, resp. module, generated by the leading monomials of the given generators of the ideal, resp. module. This is also the dimension of the ideal resp. submodule, if it is represented by a left Groebner basis.

Note:

The dimension of a submodule of a free module is defined to be the Gelfand-Kirillov dimension of the left module with the presentation via given submodule.
The computed Gelfand-Kirillov dimension is taken relative to the ground field. In order to compute the complete Gelfand-Kirillov dimension, one has to add the transcendence degree of the ground field over its prime field.

Example:

ring r=0,(x,y,Dx,Dy),dp; matrix M[4][4]; M[1,3]=1;M[2,4]=1; def R = nc_algebra(1,M); // 2nd Weyl algebra setring R; dim(std(0)); // the GK dimension of the ring itself ==> 4 ideal I=x*Dy^2-2*y*Dy^2+2*Dy, Dx^3+3*Dy^2; dim(std(I)); // the GK dimension of the module R/I ==> 2 module T = (x*Dx -2)*gen(1), Dx^3*gen(1), (y*Dy +3)*gen(2); dim(std(T)); // the GK dimension of the module R^2/T ==> 3