Singular Manual: grgens

| #### D.15.6.12 grgens Procedure from library `gradedModules.lib` (see gradedModules_lib). Usage: grgens(M), graded object M (map) Return: graded object Purpose: try compute graded generators of coker(M) and return them as columns of a graded map. Note: presentation of resulting generated submodule may be different to M! Example:
    LIB "gradedModules.lib";
    ring r=32003,(x,y,z),dp;
    module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
    ==> Graded homomorphism: r(-2) + r + r(4)^2 <- 0, given by zero (4 x 0) matri\
       x.
    module N = grgens(M);
    grview( N ); print(N); // fine == M
    ==> Graded homomorphism: r(-2) + r + r(4)^2 <- r(-2) + r + r(4)^2, given by a\
        diagonal matrix, with degrees:
    ==>      ..1 ..2 ..3 ..4 ....
    ==>      --- --- --- --- +...
    ==>   2 :  0   -   -   - |..1
    ==>   0 :  -   0   -   - |..2
    ==>  -4 :  -   -   0   - |..3
    ==>  -4 :  -   -   -   0 |..4
    ==>      === === === ===
    ==>        2   0  -4  -4
    ==> 1,0,0,0,
    ==> 0,1,0,0,
    ==> 0,0,1,0,
    ==> 0,0,0,1
    module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
    A = grgroebner(A); grview(A);
    ==> Graded homomorphism: r^3 + r(-1) <- r(-1)^3 + r(-2) + r(-3), given by a m\
       atrix, with degrees:
    ==>      ..1 ..2 ..3 ..4 ..5 ....
    ==>      --- --- --- --- --- +...
    ==>   0 :  1   1   1   2   - |..1
    ==>   0 :  1   -   1   -   - |..2
    ==>   0 :  1   1   1   2   3 |..3
    ==>   1 :  0   0   0   1   2 |..4
    ==>      === === === === ===
    ==>        1   1   1   2   3
    module B = grgens(A);
    grview( B ); print(B); // Ups :( != A
    ==> Graded homomorphism: r(2) <- r^3 + r(-1), given by a matrix, with degrees\
       :
    ==>      ..1 ..2 ..3 ..4 ....
    ==>      --- --- --- --- +...
    ==>  -2 :  2   2   2   3 |..1
    ==>      === === === ===
    ==>        0   0   0   1
    ==> xy-3y2+xz+3yz,-xy+2y2+2xz+2yz,x2-xy-4y2,y3-x2z-2xyz-y2z
    grview( grgens( grzero() ) );
    ==> Graded homomorphism: 0 <- 0, given by zero (0^2) matrix.

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