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D.15.18.29 mappingcone

Procedure from library gradedModules.lib (see gradedModules_lib).

Usage:
mappingcone(M,N), M,N graded objects

Return:
chain complex (as a list)

Purpose:
construct a free resolution of the cokernel of a random map between Img(M), and Img(N).

Example:
 
LIB "gradedModules.lib";
ring r=32003, (x(0..4)),dp;
def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
def M=grgens(A);
grview(M);
==> Graded homomorphism: r(-1)^3 <- r(-1)^3, given by a diagonal matrix, with\
    degrees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>   1 :  0   -   - |..1
==>   1 :  -   0   - |..2
==>   1 :  -   -   0 |..3
==>      === === ===     
==>        1   1   1     
def B=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
def N=grgens(B);
grview(N);
==> Graded homomorphism: r^5 <- r(-1)^10, given by a matrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==>      --- --- --- --- --- --- --- --- --- --- +...
==>   0 :  -   -   -   -   -   -   1   1   1   1 |..1
==>   0 :  1   1   -   1   -   -   1   -   -   - |..2
==>   0 :  1   -   1   -   1   -   -   1   -   - |..3
==>   0 :  -   1   1   -   -   1   -   -   1   - |..4
==>   0 :  -   -   -   1   1   1   -   -   -   1 |..5
==>      === === === === === === === === === ===     
==>        1   1   1   1   1   1   1   1   1   1     
def R=grlifting(M,N);
==> t:  2
grview(R);
==> 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     
def T=mappingcone(M,N);
==> t:  2
grview(T);
==> 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     
def U=grtranspose(T[1]);
resolution G=mres(U,0);
print(betti(G),"betti");
==>            0     1     2
==> ------------------------
==>    -2:    10     7     -
==>    -1:     -     -     -
==>     0:     -     -     1
==> ------------------------
==> total:    10     7     1
==> 
ideal I=groebner(flatten(G[2]));
resolution GI=mres(I,0);
print(betti(GI),"betti");
==>            0     1     2     3     4
==> ------------------------------------
==>     0:     1     -     -     -     -
==>     1:     -     -     -     -     -
==>     2:     -     7    10     5     1
==> ------------------------------------
==> total:     1     7    10     5     1
==>