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D.15.20.30 grlifting3
Procedure from library gradedModules.lib (see gradedModules_lib).
- Todo:
- grlifting4 was newer and had more documentation than this proc, but was removed... Please verify and update!
Example:
| | LIB "gradedModules.lib";
ring r=32003, x(0..4),dp;
def A=grtwist(3,1);
grview(A);
==> Graded homomorphism: r(1)^3 <- 0, given by zero (3 x 0) matrix.
def T=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
grview(T);
==> Graded homomorphism: r(-1)^10 <- r(-2)^10, given by a square matrix, with\
degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==> --- --- --- --- --- --- --- --- --- --- +...
==> 1 : 1 1 - - 1 - - - - - |..1
==> 1 : 1 - 1 - - 1 - - - - |..2
==> 1 : 1 - - 1 - - 1 - - - |..3
==> 1 : - 1 1 - - - - 1 - - |..4
==> 1 : - 1 - 1 - - - - 1 - |..5
==> 1 : - - 1 1 - - - - - 1 |..6
==> 1 : - - - - 1 1 - 1 - - |..7
==> 1 : - - - - 1 - 1 - 1 - |..8
==> 1 : - - - - - 1 1 - - 1 |..9
==> 1 : - - - - - - - 1 1 1 |.10
==> === === === === === === === === === ===
==> 2 2 2 2 2 2 2 2 2 2
def F=grlifting3(T,A);
==> 0 1 2 3
==> ------------------------------
==> 1: 10 10 5 1
==> ------------------------------
==> total: 10 10 5 1
==>
==> 0
==> ------------
==> -1: 3
==> ------------
==> total: 3
==>
==> t: 1
==> Graded homomorphism: r(1)^3 <- r(-1)^10, given by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==> --- --- --- --- --- --- --- --- --- --- +...
==> -1 : 2 2 2 2 2 2 2 2 2 2 |..1
==> -1 : 2 2 2 2 2 2 2 2 2 2 |..2
==> -1 : 2 2 2 2 2 2 2 2 2 2 |..3
==> === === === === === === === === === ===
==> 1 1 1 1 1 1 1 1 1 1
grview(F);
==> Graded resolution:
==> r(1)^3 <-- d_1 --
==> r(-1)^10, given by maps:
==> d_1 :
==> Graded homomorphism: r(1)^3 <- r(-1)^10, given by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==> --- --- --- --- --- --- --- --- --- --- +...
==> -1 : 2 2 2 2 2 2 2 2 2 2 |..1
==> -1 : 2 2 2 2 2 2 2 2 2 2 |..2
==> -1 : 2 2 2 2 2 2 2 2 2 2 |..3
==> === === === === === === === === === ===
==> 1 1 1 1 1 1 1 1 1 1
def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));
def H=grlifting3(R, S);
==> 0 1
==> ------------------
==> 1: 10 2
==> ------------------
==> total: 10 2
==>
==> 0 1 2 3
==> ------------------------------
==> 0: 1 - - -
==> 1: 20 20 10 2
==> ------------------------------
==> total: 21 20 10 2
==>
==> t: 2
==> Graded homomorphism: r(-2)^20 <- r(-2)^2, given by a matrix, with degrees\
:
==> ..1 ..2 ....
==> --- --- +...
==> 2 : 0 0 |..1
==> 2 : 0 0 |..2
==> 2 : 0 0 |..3
==> 2 : 0 0 |..4
==> 2 : 0 0 |..5
==> 2 : 0 0 |..6
==> 2 : 0 0 |..7
==> 2 : 0 0 |..8
==> 2 : 0 0 |..9
==> 2 : 0 0 |.10
==> 2 : 0 0 |.11
==> 2 : 0 0 |.12
==> 2 : 0 0 |.13
==> 2 : 0 0 |.14
==> 2 : 0 0 |.15
==> 2 : 0 0 |.16
==> 2 : 0 0 |.17
==> 2 : 0 0 |.18
==> 2 : 0 0 |.19
==> 2 : 0 0 |.20
==> === ===
==> 2 2
==> k: 1
==> Graded homomorphism: r + r(-1)^20 <- r(-1)^10, given by a matrix, with de\
grees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==> --- --- --- --- --- --- --- --- --- --- +...
==> 0 : - - - - - - - - - - |..1
==> 1 : 0 0 0 - - 0 0 0 - - |..2
==> 1 : 0 0 - 0 - 0 0 - 0 - |..3
==> 1 : 0 0 - - 0 0 0 - - 0 |..4
==> 1 : 0 - 0 0 - 0 - 0 0 - |..5
==> 1 : 0 - 0 - 0 0 - 0 - 0 |..6
==> 1 : 0 - - 0 0 0 - - 0 0 |..7
==> 1 : - 0 0 0 - - 0 0 0 - |..8
==> 1 : - 0 0 - 0 - 0 0 - 0 |..9
==> 1 : - 0 - 0 0 - 0 - 0 0 |.10
==> 1 : - - 0 0 0 - - 0 0 0 |.11
==> 1 : 0 0 0 - - 0 0 0 - - |.12
==> 1 : 0 0 - 0 - 0 0 - 0 - |.13
==> 1 : 0 0 - - 0 0 0 - - 0 |.14
==> 1 : 0 - 0 0 - 0 - 0 0 - |.15
==> 1 : 0 - 0 - 0 0 - 0 - 0 |.16
==> 1 : 0 - - 0 0 0 - - 0 0 |.17
==> 1 : - 0 0 0 - - 0 0 0 - |.18
==> 1 : - 0 0 - 0 - 0 0 - 0 |.19
==> 1 : - 0 - 0 0 - 0 - 0 0 |.20
==> 1 : - - 0 0 0 - - 0 0 0 |.21
==> === === === === === === === === === ===
==> 1 1 1 1 1 1 1 1 1 1
// grview(H);
// 2nd module does not lie in the first:
// def H=grlifting3(S, R);
//def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
//def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
//def N=grlifting3(I,J); grview(N);
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