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D.4.8.9 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;