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7.8.5 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
See ideal; liftstd (letterplace); syz (letterplace); twostd (letterplace).

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