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D.15.5.31 mappingcone3
Procedure from library gradedModules.lib (see gradedModules_lib).
- Usage:
- mappingcone3(A,B), graded objects A and B (matrices defining maps)
- Return:
- chain complex (as a list)
- Purpose:
- construct a free resolution of the cokernel of a random map between M=coker(A), and N=coker(B)
Example:
| | LIB "gradedModules.lib";
ring r=32003,x(0..4),dp;
def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
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(A,T); grview(F);
==> 0
==> ------------
==> 1: 3
==> ------------
==> total: 3
==>
==> 0 1 2 3
==> ------------------------------
==> 1: 10 10 5 1
==> ------------------------------
==> total: 10 10 5 1
==>
==> t: 1
==> Graded homomorphism: r(-1)^10 <- r(-1)^3, given by a matrix, with degrees\
:
==> ..1 ..2 ..3 ....
==> --- --- --- +...
==> 1 : 0 0 0 |..1
==> 1 : 0 0 0 |..2
==> 1 : 0 0 0 |..3
==> 1 : 0 0 0 |..4
==> 1 : 0 0 0 |..5
==> 1 : 0 0 0 |..6
==> 1 : 0 0 0 |..7
==> 1 : 0 0 0 |..8
==> 1 : 0 0 0 |..9
==> 1 : 0 0 0 |.10
==> === === ===
==> 1 1 1
==> Graded resolution:
==> r(-1)^10 <-- d_1 --
==> r(-1)^3, given by maps:
==> d_1 :
==> Graded homomorphism: r(-1)^10 <- r(-1)^3, given by a matrix, with degrees\
:
==> ..1 ..2 ..3 ....
==> --- --- --- +...
==> 1 : 0 0 0 |..1
==> 1 : 0 0 0 |..2
==> 1 : 0 0 0 |..3
==> 1 : 0 0 0 |..4
==> 1 : 0 0 0 |..5
==> 1 : 0 0 0 |..6
==> 1 : 0 0 0 |..7
==> 1 : 0 0 0 |..8
==> 1 : 0 0 0 |..9
==> 1 : 0 0 0 |.10
==> === === ===
==> 1 1 1
// BUG in the proc
def G=mappingcone3(A,T); grview(G);
==> 0
==> ------------
==> 1: 3
==> ------------
==> total: 3
==>
==> 0 1 2 3
==> ------------------------------
==> 1: 10 10 5 1
==> ------------------------------
==> total: 10 10 5 1
==>
==> t: 1
==> Graded homomorphism: r(-1)^10 <- r(-1)^3, given by a matrix, with degrees\
:
==> ..1 ..2 ..3 ....
==> --- --- --- +...
==> 1 : 0 0 0 |..1
==> 1 : 0 0 0 |..2
==> 1 : 0 0 0 |..3
==> 1 : 0 0 0 |..4
==> 1 : 0 0 0 |..5
==> 1 : 0 0 0 |..6
==> 1 : 0 0 0 |..7
==> 1 : 0 0 0 |..8
==> 1 : 0 0 0 |..9
==> 1 : 0 0 0 |.10
==> === === ===
==> 1 1 1
==> Graded resolution:
==> r(-1)^10 <-- d_1 --
==> r(-1)^3 + r(-2)^10, given by maps:
==> d_1 :
==> Graded homomorphism: r(-1)^10 <- r(-1)^3 + r(-2)^10, given by a matrix, w\
ith degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 ....
==> --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==> 1 : 0 0 0 1 1 - - 1 - - - - - |..1
==> 1 : 0 0 0 1 - 1 - - 1 - - - - |..2
==> 1 : 0 0 0 1 - - 1 - - 1 - - - |..3
==> 1 : 0 0 0 - 1 1 - - - - 1 - - |..4
==> 1 : 0 0 0 - 1 - 1 - - - - 1 - |..5
==> 1 : 0 0 0 - - 1 1 - - - - - 1 |..6
==> 1 : 0 0 0 - - - - 1 1 - 1 - - |..7
==> 1 : 0 0 0 - - - - 1 - 1 - 1 - |..8
==> 1 : 0 0 0 - - - - - 1 1 - - 1 |..9
==> 1 : 0 0 0 - - - - - - - 1 1 1 |.10
==> === === === === === === === === === === === === ===
==> 1 1 1 2 2 2 2 2 2 2 2 2 2
/*
module W=grtranspose(G[1]);
resolution U=mres(W,0);
print(betti(U,0),"betti"); // ?
ideal P=groebner(flatten(U[2]));
resolution L=mres(P,0);
print(betti(L),"betti");
*/
def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
grview(R);
==> Graded homomorphism: r(-1)^10 <- r(-2)^2, given by a matrix, with degrees\
:
==> ..1 ..2 ....
==> --- --- +...
==> 1 : 1 - |..1
==> 1 : 1 - |..2
==> 1 : 1 - |..3
==> 1 : 1 - |..4
==> 1 : 1 - |..5
==> 1 : - 1 |..6
==> 1 : - 1 |..7
==> 1 : - 1 |..8
==> 1 : - 1 |..9
==> 1 : - 1 |.10
==> === ===
==> 2 2
def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));
grview(S);
==> Graded homomorphism: r + r(-1)^20 <- r(-2)^20, given by a matrix, with de\
grees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 .14 .15 .16 .17 \
.18 .19 .20 ....
==> --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==> 0 : - - - - - - - - - - - - - - - - - \
- - - |..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
==> 1 : - - - - - - - 1 1 1 - - - - - - - \
- - - |.11
==> 1 : - - - - - - - - - - 1 1 - - 1 - - \
- - - |.12
==> 1 : - - - - - - - - - - 1 - 1 - - 1 - \
- - - |.13
==> 1 : - - - - - - - - - - 1 - - 1 - - 1 \
- - - |.14
==> 1 : - - - - - - - - - - - 1 1 - - - - \
1 - - |.15
==> 1 : - - - - - - - - - - - 1 - 1 - - - \
- 1 - |.16
==> 1 : - - - - - - - - - - - - 1 1 - - - \
- - 1 |.17
==> 1 : - - - - - - - - - - - - - - 1 1 - \
1 - - |.18
==> 1 : - - - - - - - - - - - - - - 1 - 1 \
- 1 - |.19
==> 1 : - - - - - - - - - - - - - - - 1 1 \
- - 1 |.20
==> 1 : - - - - - - - - - - - - - - - - - \
1 1 1 |.21
==> === === === === === === === === === === === === === === === === === === === ===
==> 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 \
2 2 2
def H=grlifting3(R,S); grview(H);
==> 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
==> Graded-object collection, given by the following maps (named here as o_[1\
.. 2]):
==> o_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
==> o_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
// BUG in the proc
def G=mappingcone3(R,S);
==> // ** redefining G (def G=mappingcone3(R,S);) ./examples/mappingcone3.sin\
g:24
==> 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
==> // ** redefining A ( module A=grconcat(P[i],rN[i]);) gradedModules.lib\
::mappingcone3:2370
==> // ** redefining B ( module B=grobj(zero,v,w);) gradedModules.lib::map\
pingcone3:2371
def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
// def N=grlifting3(I,J);
// 2nd module does not lie in the first:
// def NN=mappingcone3(I,J); // ????????
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