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D.4.3.14 kernel
Procedure from library homolog.lib (see homolog_lib).
- Usage:
- kernel(A,M,N);
- Compute:
- Let M and N be submodules of R^m and R^n, presenting M'=R^m/M,
N'=R^n/N (R=basering), and let A:R^m-->R^n be a matrix inducing a
map A':M'-->N'. Then kernel(A,M,N); computes a presentation K of
ker(A') as in the commutative diagram:
| ker(A') ---> M' --A'--> N'
|^ |^ |^
| | |
R^r ---> R^m --A--> R^n
|^ |^ |^
|K |M |N
| | |
R^s ---> R^p -----> R^q
|
- Return:
- module K, a presentation of ker(A':coker(M)->coker(N)).
Example:
| LIB "homolog.lib";
ring r;
module N=[2x,x],[0,y];
module M=maxideal(1)*freemodule(2);
matrix A[2][2]=2x,0,x,y,z2,y;
module K=kernel(A,M,N);
// dimension of kernel:
dim(std(K));
==> 0
// vector space dimension of kernel:
vdim(std(K));
==> 2
print(K);
==> z,0,y,0,x,0,
==> 0,z,0,y,0,x
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