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