5.1.66 lift

Syntax:

lift ( ideal_expression, subideal_expression )
lift ( module_expression, submodule_expression )
lift ( ideal_expression, subideal_expression, matrix_name )
lift ( module_expression, submodule_expression, matrix_name )

Type:

matrix

Purpose:

computes the transformation matrix which expresses the generators of a submodule in terms of the generators of a module. Uses different algorithms for modules which are, resp. are not, represented by a standard basis.
More precisely, if m is the module (or ideal), sm the submodule (or ideal), and T the transformation matrix returned by lift, then matrix(sm)*U = matrix(m)*T and module(sm*U) = module(matrix(m)*T) (resp. ideal(sm*U) = ideal(matrix(m)*T)), where U is a diagonal matrix of units.
U is always the unity matrix if the basering is a polynomial ring (not power series ring). U is stored in the optional third argument.

Note:

Gives a warning if sm is not a submodule.

Example:

ring r=32003,(x,y,z),(dp,C); ideal m=3x2+yz,7y6+2x2y+5xz; poly f=y7+x3+xyz+z2; ideal i=jacob(f); matrix T=lift(i,m); matrix(m)-matrix(i)*T; → _[1,1]=0 → _[1,2]=0

See division; ideal; module.