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D.15.5.29 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
==>
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