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D.15.10.12 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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