  
  [1X4 Example[0X
  
  
  [1X4.1 HomHom[0X
  
  The following example is taken from Section 2 of [BR06].
  The  computation  takes  place  over  the  ring  R=ℤ/2^8ℤ, which is directly
  supported by the package [5XGauss[0m.
  
  Here we compute the (infinite) long exact homology sequence of the covariant
  functor Hom(Hom(-,ℤ/2^7ℤ),ℤ/2^4ℤ) (and its left derived functors) applied to
  the short exact sequence
  0 -> M_=ℤ/2^2ℤ --alpha_1--> M=ℤ/2^5ℤ --alpha_2--> _M=ℤ/2^3ℤ -> 0.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> R := HomalgRingOfIntegers( 2^8 );[0X
    [4X<A homalg internal ring>[0X
    [4Xgap> Display( R );[0X
    [4XZ/256Z[0X
    [4Xgap> M := LeftPresentation( [ 2^5 ], R );[0X
    [4X<A cyclic left module presented by an unknown number of relations for a cyclic\[0X
    [4X generator>[0X
    [4Xgap> Display( M );[0X
    [4XZ/256Z/< ZmodnZObj(32,256) >[0X
    [4Xgap> M;[0X
    [4X<A cyclic left module presented by 1 relation for a cyclic generator>[0X
    [4Xgap> _M := LeftPresentation( [ 2^3 ], R );[0X
    [4X<A cyclic left module presented by an unknown number of relations for a cyclic\[0X
    [4X generator>[0X
    [4Xgap> Display( _M );[0X
    [4XZ/256Z/< ZmodnZObj(8,256) >[0X
    [4Xgap> _M;[0X
    [4X<A cyclic left module presented by 1 relation for a cyclic generator>[0X
    [4Xgap> alpha2 := HomalgMap( [ 1 ], M, _M );[0X
    [4X<A "homomorphism" of left modules>[0X
    [4Xgap> IsMorphism( alpha2 );[0X
    [4Xtrue[0X
    [4Xgap> alpha2;[0X
    [4X<A homomorphism of left modules>[0X
    [4Xgap> Display( alpha2 );[0X
    [4X   1[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4Xgap> M_ := Kernel( alpha2 );[0X
    [4X<A cyclic left module presented by yet unknown relations for a cyclic generato\[0X
    [4Xr>[0X
    [4Xgap> alpha1 := KernelEmb( alpha2 );[0X
    [4X<A monomorphism of left modules>[0X
    [4Xgap> seq := HomalgComplex( alpha2 );[0X
    [4X<An acyclic complex containing a single morphism of left modules at degrees [0X
    [4X[ 0 .. 1 ]>[0X
    [4Xgap> Add( seq, alpha1 );[0X
    [4Xgap> seq;[0X
    [4X<A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]>[0X
    [4Xgap> IsShortExactSequence( seq );[0X
    [4Xtrue[0X
    [4Xgap> seq;[0X
    [4X<A short exact sequence containing 2 morphisms of left modules at degrees [0X
    [4X[ 0 .. 2 ]>[0X
    [4Xgap> Display( seq );[0X
    [4X-------------------------[0X
    [4Xat homology degree: 2[0X
    [4XZ/256Z/< ZmodnZObj(4,256) > [0X
    [4X-------------------------[0X
    [4X  24[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 1[0X
    [4XZ/256Z/< ZmodnZObj(32,256) > [0X
    [4X-------------------------[0X
    [4X   1[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 0[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4Xgap> K := LeftPresentation( [ 2^7 ], R );[0X
    [4X<A cyclic left module presented by an unknown number of relations for a cyclic\[0X
    [4X generator>[0X
    [4Xgap> L := RightPresentation( [ 2^4 ], R );[0X
    [4X<A cyclic right module on a cyclic generator satisfying an unknown number of r\[0X
    [4Xelations>[0X
    [4Xgap> triangle := LHomHom( 4, seq, K, L, "t" );[0X
    [4X<An exact triangle containing 3 morphisms of left complexes at degrees [0X
    [4X[ 1, 2, 3, 1 ]>[0X
    [4Xgap> lehs := LongSequence( triangle );[0X
    [4X<A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]>[0X
    [4Xgap> ByASmallerPresentation( lehs );[0X
    [4X<A non-zero sequence containing 14 morphisms of left modules at degrees [0X
    [4X[ 0 .. 14 ]>[0X
    [4Xgap> IsExactSequence( lehs );[0X
    [4Xfalse[0X
    [4Xgap> lehs;[0X
    [4X<A non-zero left acyclic complex containing [0X
    [4X14 morphisms of left modules at degrees [ 0 .. 14 ]>[0X
    [4Xgap> Assert( 0, IsLeftAcyclic( lehs ) );[0X
    [4Xgap> Display( lehs );[0X
    [4X-------------------------[0X
    [4Xat homology degree: 14[0X
    [4XZ/256Z/< ZmodnZObj(4,256) > [0X
    [4X-------------------------[0X
    [4X   4[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 13[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   6[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 12[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   2[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 11[0X
    [4XZ/256Z/< ZmodnZObj(4,256) > [0X
    [4X-------------------------[0X
    [4X   4[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 10[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   6[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 9[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   2[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 8[0X
    [4XZ/256Z/< ZmodnZObj(4,256) > [0X
    [4X-------------------------[0X
    [4X   4[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 7[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   6[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 6[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   2[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 5[0X
    [4XZ/256Z/< ZmodnZObj(4,256) > [0X
    [4X-------------------------[0X
    [4X   4[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 4[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   6[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 3[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
    [4X   2[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 2[0X
    [4XZ/256Z/< ZmodnZObj(4,256) > [0X
    [4X-------------------------[0X
    [4X   8[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 1[0X
    [4XZ/256Z/< ZmodnZObj(16,256) > [0X
    [4X-------------------------[0X
    [4X   1[0X
    [4X[0X
    [4Xthe map is currently represented by the above 1 x 1 matrix[0X
    [4X------------v------------[0X
    [4Xat homology degree: 0[0X
    [4XZ/256Z/< ZmodnZObj(8,256) > [0X
    [4X-------------------------[0X
  [4X------------------------------------------------------------------[0X
  
