D.4.3.9 homology

Procedure from library homolog.lib (see homolog_lib).

Usage:

homology(A,B,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,B matrices inducing maps

R^k --A--> R^m --B--> R^n.

Compute a presentation of the module

ker(B)/im(A) := ker(M'/im(A) --B--> N'/im(BM)+im(BA)).

If B induces a map M’–>N’ (i.e BM=0) and if im(A) is contained in ker(B) (that is, BA=0) then ker(B)/im(A) is the homology of the complex

R^k--A-->M'--B-->N'.

Return:

module H, a presentation of ker(B)/im(A).

Note:

homology returns a free module of rank m if ker(B)=im(A).

Example:

LIB "homolog.lib";
ring r;
ideal id=maxideal(4);
qring qr=std(id);
module N=maxideal(3)*freemodule(2);
module M=maxideal(2)*freemodule(2);
module B=[2x,0],[x,y],[z2,y];
module A=M;
module H=homology(A,B,M,N);
H=std(H);
// dimension of homology:
dim(H);
→ 0
// vector space dimension: 
vdim(H);
→ 19
ring s=0,x,ds;
qring qs=std(x4);
module A=[x];
module B=A;
module M=[x3];
module N=M;
homology(A,B,M,N);
→ _[1]=gen(1)