Singular

7.8.2 lift (letterplace)

Syntax:

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

Type:

matrix

Purpose:

computes the transformation matrix which expresses the generators of a subbimodule in terms of the generators of a bimodule.
More precisely, if m is the module (or ideal), sm the submodule (or ideal), and T the transformation matrix returned by lift, then the substitution of each ncgen(i) in T by the m[i] delivers a matrix, say N. The i-th generator of sm is equal to the sum of elements in the i-th column of N.

Note:

Gives a warning if sm is not a submodule.

Note:

The procedure testLift can be used for testing the result.

Example:

LIB "freegb.lib"; ring r = 0,(x,y),(c,Dp); ring R = freeAlgebra(r, 7, 2); ideal I = std(x*y*x + 1); print(matrix(I)); ==> x*y-y*x,y*x*x+1 ideal SI = x*I[1]*y + y*x*I[2], I[1]*y*x + I[2]*y; matrix T = lift(I, SI); print(T); ==> y*ncgen(1)*x*x+x*ncgen(1)*y,y*x*ncgen(1)+y*ncgen(1)*x+ncgen(1)*y*x, ==> y*ncgen(2)*x, y*ncgen(2) print(matrix(SI)); // the original generators ==> y*x*y*x*x+x*x*y*y-x*y*x*y+y*x,x*y*y*x+y*x*x*y-y*x*y*x+y print(matrix(testLift(I,T))); // test for the result of lift ==> y*x*y*x*x+x*x*y*y-x*y*x*y+y*x,x*y*y*x+y*x*x*y-y*x*y*x+y