#### D.4.11.9 Hom

Procedure from library `homolog.lib` (see homolog_lib).

Usage:
  Hom(M,N,[any]); M,N=modules

Compute:
  A presentation of Hom(M',N'), M'=coker(M), N'=coker(N) as follows: let

         F1 --M-> F0 -->M' --> 0,    G1 --N-> G0 --> N' --> 0

  be presentations of M' and N'. Consider

                                        0               0
                                        |^              |^
             0 --> Hom(M',N') ----> Hom(F0,N') ----> Hom(F1,N')
                                        |^              |^
        (A:  induced by M)          Hom(F0,G0) --A-> Hom(F1,G0)
                                        |^              |^
        (B,C:induced by N)              |C              |B
                                    Hom(F0,G1) ----> Hom(F1,G1)

  Let D=modulo(A,B) and Hom=modulo(D,C), then we have exact sequences

         R^p  --D-> Hom(F0,G0) --A-> Hom(F1,G0)/im(B),

       R^q -Hom-> R^p --D-> Hom(F0,G0)/im(C) --A-> Hom(F1,G0)/im(B).

  Hence Hom presents Hom(M',N')

Return:
  module Hom, a presentation of Hom(M',N'), resp., in case of 3 arguments, a list l (of size <=3):

                 - l[1] = Hom
                 - l[2] = SB of Hom
                 - l[3] = kbase of coker(Hom) (if finite dimensional, not 0),
                          represented by elements in Hom(F0,G0) via mapping D

Display:
  printlevel >=0: (affine) dimension of Hom (default)
  printlevel >=1: D and C and kbase of coker(Hom) in Hom(F0,G0)
  printlevel >=2: elements of kbase of coker(Hom) as matrix :F0-->G0

Note:
  DISPLAY is as described only for a direct call of 'Hom'. Calling 'Hom' from another proc has the same effect as decreasing printlevel by 1.

Example:

    LIB "homolog.lib";
    int p     = printlevel;
    printlevel= 1;   //in 'example proc' printlevel has to be increased by 1
    ring r    = 0,(x,y),dp;
    ideal i   = x2-y3,xy;
    qring q   = std(i);
    ideal i   = fetch(r,i);
    module M  = [-x,y],[-y2,x],[x3];
    module H  = Hom(M,i);
    ==> // dimension of Hom:  0
    ==> // vdim of Hom:       5
    ==>
    ==> // given  F1 --M-> F0 -->M'--> 0 and  G1 --N-> G0 -->N'--> 0,
    ==> // show D = ker( Hom(F0,G0) --> Hom(F1,G0)/im(Hom(F1,G1)->Hom(F1,G0)) )
    ==> y,x, 0,
    ==> x,y2,x2
    ==> // show C = im ( Hom(F0,G1) --> Hom(F0,G0) )
    ==> -y3+x2,0,     xy,0,
    ==> 0,     -y3+x2,0, xy
    ==>
    print(H);
    ==> 0, x, 0,y2,0,
    ==> y, 0, 0,-x,x2,
    ==> -1,-1,x,0, 0
    printlevel= 2;
    list L    = Hom(M,i,1);"";
    ==> // dimension of Hom:  0
    ==> // vdim of Hom:       5
    ==>
    ==> // given  F1 --M-> F0 -->M'--> 0 and  G1 --N-> G0 -->N'--> 0,
    ==> // show D = ker( Hom(F0,G0) --> Hom(F1,G0)/im(Hom(F1,G1)->Hom(F1,G0)) )
    ==> y,x, 0,
    ==> x,y2,x2
    ==> // show C = im ( Hom(F0,G1) --> Hom(F0,G0) )
    ==> -y3+x2,0,     xy,0,
    ==> 0,     -y3+x2,0, xy
    ==>
    ==> // element 1 of kbase of Hom in Hom(F0,G0) as matrix: F0-->G0:
    ==> y2,xy
    ==> // element 2 of kbase of Hom in Hom(F0,G0) as matrix: F0-->G0:
    ==> y,x
    ==> // element 3 of kbase of Hom in Hom(F0,G0) as matrix: F0-->G0:
    ==> x2,xy2
    ==> // element 4 of kbase of Hom in Hom(F0,G0) as matrix: F0-->G0:
    ==> x,y2
    ==> // element 5 of kbase of Hom in Hom(F0,G0) as matrix: F0-->G0:
    ==> 0,x2
    ==>
    printlevel=1;
    ring s    = 3,(x,y,z),(c,dp);
    ideal i   = jacob(ideal(x2+y5+z4));
    qring rq=std(i);
    matrix M[2][2]=xy,x3,5y,4z,x2;
    matrix N[3][2]=x2,x,y3,3xz,x2z,z;
    print(M);
    ==> xy,x3,
    ==> -y,z
    print(N);
    ==> x2, x,
    ==> y3, 0,
    ==> x2z,z
    list l=Hom(M,N,1);
    ==> // dimension of Hom:  0
    ==> // vdim of Hom:       16
    ==>
    ==> // given  F1 --M-> F0 -->M'--> 0 and  G1 --N-> G0 -->N'--> 0,
    ==> // show D = ker( Hom(F0,G0) --> Hom(F1,G0)/im(Hom(F1,G1)->Hom(F1,G0)) )
    ==> 0,0, 0,0, 0,   0,0,   1,
    ==> 0,0, 0,0, 0,   0,y3z2,0,
    ==> 0,0, 0,0, 0,   1,0,   0,
    ==> 0,0, 0,y3,y2z2,0,0,   0,
    ==> 0,0, 1,0, 0,   0,0,   0,
    ==> z,y3,0,0, 0,   0,0,   0
    ==> // show C = im ( Hom(F0,G1) --> Hom(F0,G0) )
    ==> x2, 0,  x,0,
    ==> 0,  x2, 0,x,
    ==> y3, 0,  0,0,
    ==> 0,  y3, 0,0,
    ==> x2z,0,  z,0,
    ==> 0,  x2z,0,z
    ==>
    ==> // columns of matrix are kbase of Hom in Hom(F0,G0)
    ==> 0, 0, 0, 0,0,0,   0,   0,  0, 0,  0, 0,0, 0,0,0,
    ==> 0, 0, 0, 0,0,0,   0,   0,  0, 0,  0, 0,0, 0,0,y3z2,
    ==> 0, 0, 0, 0,0,0,   y2z2,yz2,z2,y2z,yz,z,y2,y,1,0,
    ==> 0, 0, 0, 0,0,y2z2,0,   0,  0, 0,  0, 0,0, 0,0,0,
    ==> 0, y3,y2,y,1,0,   0,   0,  0, 0,  0, 0,0, 0,0,0,
    ==> y3,0, 0, 0,0,0,   0,   0,  0, 0,  0, 0,0, 0,0,0
    printlevel = p;

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