### 5.1.75 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. 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.
`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.

User manual for Singular version 4-0-3, 2016, generated by texi2html.