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D.15.8.19 grres
Procedure from library gradedModules.lib (see gradedModules_lib).
- Usage:
- grres(M, l[, b]), graded object M, int l, int b
- Return:
- graded resolution = list of graded objects
- Purpose:
- compute graded resolution of M (of length l) and minimise it if b was given
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
module B = grgroebner(A);
grview(B);
==> 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 resolution of B: "; def C = grres(B, 0); grview(C);
==> graded resolution of B:
==> Graded resolution:
==> r^3 + r(-1) <-- d_1 --
==> r(-1) + r(-2) + r(-1)^2 + r(-3) <-- d_2 --
==> r(-3) + r(-2) <-- d_3 --
==> 0, given by maps:
==> d_1 :
==> Graded homomorphism: r^3 + r(-1) <- r(-1) + r(-2) + r(-1)^2 + r(-3), give\
n by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ....
==> --- --- --- --- --- +...
==> 0 : 1 2 1 1 - |..1
==> 0 : - - 1 1 - |..2
==> 0 : 1 2 1 1 3 |..3
==> 1 : 0 1 0 0 2 |..4
==> === === === === ===
==> 1 2 1 1 3
==> d_2 :
==> Graded homomorphism: r(-1) + r(-2) + r(-1)^2 + r(-3) <- r(-3) + r(-2), gi\
ven by a matrix, with degrees:
==> ..1 ..2 ....
==> --- --- +...
==> 1 : 2 1 |..1
==> 2 : 1 0 |..2
==> 1 : - 1 |..3
==> 1 : - 1 |..4
==> 3 : 0 - |..5
==> === ===
==> 3 2
==> d_3 :
==> Graded homomorphism: r(-3) + r(-2) <- 0, given by zero (2 x 0) matrix.
int i; int l = size(C);
"D^2 == 0: "; for (i = 1; i < l; i++ ) { i; grview( grprod(C[i], C[i+1]) ); }
==> D^2 == 0:
==> 1
==> Graded homomorphism: r^3 + r(-1) <- r(-3) + r(-2), given by zero (4 x 2) \
matrix.
==> 2
==> Graded homomorphism: r(-1) + r(-2) + r(-1)^2 + r(-3) <- 0, given by zero \
(5 x 0) matrix.
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