LIB "gradedModules.lib";
ring r=32003,(x,y,z),dp;
def A = grgroebner( grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) ) );
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     
"graded transpose: "; def B = grtranspose(A); grview( B ); print(B);
==> graded transpose: 
==> Graded homomorphism: r(1)^3 + r(2) + r(3) <- r^3 + r(1), given by a matri\
   x, with degrees: 
==>      ..1 ..2 ..3 ..4 ....
==>      --- --- --- --- +...
==>  -1 :  1   1   1   0 |..1
==>  -1 :  1   -   1   0 |..2
==>  -1 :  1   1   1   0 |..3
==>  -2 :  2   -   2   1 |..4
==>  -3 :  -   -   3   2 |..5
==>      === === === ===     
==>        0   0   0  -1     
==> y,   y,z,              1,        
==> x+2y,0,-y+z,           2,        
==> -y,  x,y-z,            1,        
==> y2,  0,-xz,            -x+3y,    
==> 0,   0,y3-x2z-2xyz-y2z,-x2+xy+4y2
"... syzygy: "; def C = grsyz(B); grview(C);
==> ... syzygy: 
==> Graded homomorphism: r^3 + r(1) <- r(-2), given by a matrix, with degrees\
   : 
==>      ..1 ....
==>      --- +...
==>   0 :  2 |..1
==>   0 :  2 |..2
==>   0 :  2 |..3
==>  -1 :  3 |..4
==>      ===     
==>        2     
"... transposed: "; def D = grtranspose(C); grview( D ); print (D);
==> ... transposed: 
==> 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
"... and back to presentation: "; def E = grsyz( D ); grview(E); print(E);
==> ... and back to presentation: 
==> Graded homomorphism: r^3 + r(-1) <- r(-1)^3, given by a matrix, with degr\
   ees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>   0 :  1   1   1 |..1
==>   0 :  1   1   1 |..2
==>   0 :  1   1   1 |..3
==>   1 :  0   -   - |..4
==>      === === ===     
==>        1   1   1     
==> y,x,   x-2y,
==> y,-2y, x-3y,
==> z,-y-z,-3z, 
==> 1,0,   0    
def F = grgens( E ); grview(F); print(F);
==> 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
def G = grpres( F ); grview(G); print(G);
==> Graded homomorphism: r^3 + r(-1) <- r(-1)^3, given by a matrix, with degr\
   ees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>   0 :  1   1   1 |..1
==>   0 :  1   1   1 |..2
==>   0 :  1   1   1 |..3
==>   1 :  0   -   - |..4
==>      === === ===     
==>        1   1   1     
==> y,x,   x-2y,
==> y,-2y, x-3y,
==> z,-y-z,-3z, 
==> 1,0,   0    
def 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.
def N = grgens(M); grview( N ); print(N);
==> 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 
def L = grpres( N ); grview( L ); print(L);
==> Graded homomorphism: r(-2) + r + r(4)^2 <- 0, given by zero (4 x 0) matri\
   x.
==> 4 x 0 zero matrix