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5.1.75 lift

lift ( ideal_expression, subideal_expression )
lift ( module_expression, submodule_expression )
lift ( ideal_expression, subideal_expression, matrix_name )
lift ( module_expression, submodule_expression, matrix_name )
computes the transformation matrix which expresses the generators of a submodule in terms of the generators of a module. Depending on which algorithm is used, modules are represented by a standard basis, or not.
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) = ideal(matrix(m)*T)), where U is a diagonal matrix of units.
U is always the identity if the basering is a polynomial ring (not power series ring). U is stored in the optional third argument.
Gives a warning if sm is not a submodule.
  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);
==> _[1,1]=0
==> _[1,2]=0
See division; ideal; module.