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D.15.20.13 grpres
Procedure from library gradedModules.lib (see gradedModules_lib).
- Usage:
- grpres(M), graded object M (submodule gens)
- Return:
- graded module (via coker)
- Purpose:
- compute graded presentation matrix of submodule generated by columns of M
Example:
| 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
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