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D.15.6.21 grprod
Procedure from library gradedModules.lib (see gradedModules_lib).
- Usage:
- grprod(M, N), graded objects M and N
- Return:
- graded object
- Purpose:
- compute graded product M * N (as composition of maps)
Example:
| LIB "gradedModules.lib";
ring r=32003,(x,y,z),dp;
module A = 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, given by a matrix, with degr\
ees:
==> ..1 ..2 ..3 ....
==> --- --- --- +...
==> 0 : 1 - 1 |..1
==> 0 : 1 1 1 |..2
==> 0 : - 1 1 |..3
==> 1 : 0 0 0 |..4
==> === === ===
==> 1 1 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 = grsyz(A);
grview(B);
==> Graded homomorphism: r(-1)^3 + r(-2) + r(-3) <- r(-2) + r(-3), given by a\
matrix, with degrees:
==> ..1 ..2 ....
==> --- --- +...
==> 1 : 1 - |..1
==> 1 : 1 2 |..2
==> 1 : 1 - |..3
==> 2 : 0 1 |..4
==> 3 : - 0 |..5
==> === ===
==> 2 3
print(B);
==> x, 0,
==> -y,y2,
==> -y,0,
==> 1, -x-2y,
==> 0, 1
module D = grprod( A, B );
grview(D);
==> Graded homomorphism: r^3 + r(-1) <- r(-2) + r(-3), given by zero (4 x 2) \
matrix.
print(D); // must be all zeroes due to syzygy property!
==> 0,0,
==> 0,0,
==> 0,0,
==> 0,0
ASSUME(0, size(D) == 0);
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