7.8.6 liftstd (letterplace)

Syntax:

liftstd ( ideal_expression, matrix_name )
liftstd ( module_expression, matrix_name )
liftstd ( ideal_expression, matrix_name, module_name )
liftstd ( module_expression, matrix_name, module_name )

Type:

ideal or module

Purpose:

returns a Groebner basis of a two-sided ideal or a bimodule and the transformation matrix from the given ideal, resp. module, to the Groebner basis from the output.
That is, 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.
In an optional third argument the syzygy bimodule will be returned.

Example:

LIB "freegb.lib"; ring r = 0,(x,y),(c,Dp); ring R = freeAlgebra(r, 8, 2); ideal I = x*y*x + 1; matrix T; module S; ideal SI = liftstd(I,T,S); print(matrix(SI)); ==> x*y-y*x,y*x*x+1 print(matrix(testLift(I,T))); // test for the result of lift ==> x*y-y*x,y*x*x+1 S; // the bisyzygy module of I ==> S[1]=[x*y*ncgen(1)*x*y+y*x*x*y*ncgen(1)-y*x*ncgen(1)*y*x-ncgen(1)*y*x*x*y\ +y*ncgen(1)-ncgen(1)*y] ==> S[2]=[x*x*y*ncgen(1)*x-x*ncgen(1)*y*x*x-x*ncgen(1)+ncgen(1)*x] ==> S[3]=[x*y*ncgen(1)*x*x*y+y*x*x*x*y*ncgen(1)-y*x*x*ncgen(1)*y*x-ncgen(1)*y\ *x*x*x*y+x*y*ncgen(1)+y*x*ncgen(1)-ncgen(1)*x*y-ncgen(1)*y*x] ==> S[4]=[x*y*x*y*ncgen(1)*x-ncgen(1)*y*x*y*x*x+y*ncgen(1)*x-ncgen(1)*y*x] testSyz(I,S); ==> _[1]=0 ==> _[2]=0 ==> _[3]=0 ==> _[4]=0