Singular Manual: homology

  | #### D.4.11.10 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); // GB
    qring qr=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)

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