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D.15.8.9 grsum
Procedure from library gradedModules.lib (see gradedModules_lib).
- Usage:
- grsum(A, B), graded objects A and B
- Return:
- graded direct sum of input objects
- Purpose:
- compute the graded direct sum of A and B
Example:
| LIB "gradedModules.lib";
// if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5;
ring r=32003,(x,y,z),dp;
module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
grview(A);
==> Graded homomorphism: r^3 + r(-1) <- r(-1)^2, given by a matrix, with degr\
ees:
==> ..1 ..2 ....
==> --- --- +...
==> 0 : 1 - |..1
==> 0 : 1 1 |..2
==> 0 : - 1 |..3
==> 1 : - - |..4
==> === ===
==> 1 1
module B = grobj( module([0,x,y]), intvec(15,1,1) );
grview(B);
==> Graded homomorphism: r(-15) + r(-1)^2 <- r(-2), given by a matrix, with d\
egrees:
==> ..1 ....
==> --- +...
==> 15 : - |..1
==> 1 : 1 |..2
==> 1 : 1 |..3
==> ===
==> 2
module C = grsum(A,B);
print(C);
==> x+y,0, 0,
==> x, x+y,0,
==> 0, y, 0,
==> 0, 0, 0,
==> 0, 0, 0,
==> 0, 0, x,
==> 0, 0, y
homog(C);
==> 1
grview(C);
==> Graded homomorphism: r^3 + r(-1) + r(-15) + r(-1)^2 <- r(-1)^2 + r(-2), g\
iven by a matrix, with degrees:
==> ..1 ..2 ..3 ....
==> --- --- --- +...
==> 0 : 1 - - |..1
==> 0 : 1 1 - |..2
==> 0 : - 1 - |..3
==> 1 : - - - |..4
==> 15 : - - - |..5
==> 1 : - - 1 |..6
==> 1 : - - 1 |..7
==> === === ===
==> 1 1 2
module D = grsum(
grsum(grpower(A,2), grtwist(1,1)),
grsum(grtwist(1,2), grpower(B,2))
);
print(D);
==> x+y,0, 0, 0, 0,0,
==> x, x+y,0, 0, 0,0,
==> 0, y, 0, 0, 0,0,
==> 0, 0, 0, 0, 0,0,
==> 0, 0, x+y,0, 0,0,
==> 0, 0, x, x+y,0,0,
==> 0, 0, 0, y, 0,0,
==> 0, 0, 0, 0, 0,0,
==> 0, 0, 0, 0, 0,0,
==> 0, 0, 0, 0, 0,0,
==> 0, 0, 0, 0, 0,0,
==> 0, 0, 0, 0, x,0,
==> 0, 0, 0, 0, y,0,
==> 0, 0, 0, 0, 0,0,
==> 0, 0, 0, 0, 0,x,
==> 0, 0, 0, 0, 0,y
homog(D);
==> 1
grview(D);
==> Graded homomorphism:
==> r^3 + r(-1) + r^3 + r(-1) + r(1) + r(2) + r(-15) + r(-1)^2 + r(-15) + r(-\
1)^2 <-
==> r(-1)^4 + r(-2)^2, given by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ....
==> --- --- --- --- --- --- +...
==> 0 : 1 - - - - - |..1
==> 0 : 1 1 - - - - |..2
==> 0 : - 1 - - - - |..3
==> 1 : - - - - - - |..4
==> 0 : - - 1 - - - |..5
==> 0 : - - 1 1 - - |..6
==> 0 : - - - 1 - - |..7
==> 1 : - - - - - - |..8
==> -1 : - - - - - - |..9
==> -2 : - - - - - - |.10
==> 15 : - - - - - - |.11
==> 1 : - - - - 1 - |.12
==> 1 : - - - - 1 - |.13
==> 15 : - - - - - - |.14
==> 1 : - - - - - 1 |.15
==> 1 : - - - - - 1 |.16
==> === === === === === ===
==> 1 1 1 1 2 2
module F = grobj( module([x,y,0]), intvec(1,1,5) );
grview(F);
==> Graded homomorphism: r(-1)^2 + r(-5) <- r(-2), given by a matrix, with de\
grees:
==> ..1 ....
==> --- +...
==> 1 : 1 |..1
==> 1 : 1 |..2
==> 5 : - |..3
==> ===
==> 2
module T = grsum( F, grsum( grtwist(1, 10), B ) );
grview(T);
==> Graded homomorphism: r(-1)^2 + r(-5) + r(10) + r(-15) + r(-1)^2 <- r(-2)^\
2
==> , given by a matrix, with degrees:
==> ...1 ...2 .....
==> ---- ---- +....
==> 1 : 1 - |...1
==> 1 : 1 - |...2
==> 5 : - - |...3
==> -10 : - - |...4
==> 15 : - - |...5
==> 1 : - 1 |...6
==> 1 : - 1 |...7
==> ==== ====
==> 2 2
// if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL
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