 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,yz, 1,
==> y2, 0,xz, x+3y,
==> 0, 0,y3x2z2xyzy2z,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
==> xy3y2+xz+3yz,xy+2y2+2xz+2yz,x2xy4y2,y3x2z2xyzy2z
"... 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, x2y,
==> y,2y, x3y,
==> z,yz,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
==> xy3y2+xz+3yz,xy+2y2+2xz+2yz,x2xy4y2,y3x2z2xyzy2z
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, x2y,
==> y,2y, x3y,
==> z,yz,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
