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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); // ????????