GAP (GaussForHomalg) - Chapter 4: Example

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4 Example
4.1 HomHom

### 4 Example

#### 4.1 HomHom

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 Gauss.

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
\(0 -> M_=ℤ/2^2ℤ --alpha_1--> M=ℤ/2^5ℤ --alpha_2--> \_M=ℤ/2^3ℤ -> 0\).

    gap> LoadPackage( "Modules", ">= 2023.10-01" );
    true
    gap> R := HomalgRingOfIntegers( 2^8 );
    Z/256Z
    gap> Display( R );
    <An internal ring>
    gap> M := LeftPresentation( [ 2^5 ], R );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> Display( M );
    Z/256Z/< ZmodnZObj(32,256) >
    gap> M;
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> _M := LeftPresentation( [ 2^3 ], R );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> Display( _M );
    Z/256Z/< ZmodnZObj(8,256) >
    gap> _M;
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> alpha2 := HomalgMap( [ 1 ], M, _M );
    <A "homomorphism" of left modules>
    gap> IsMorphism( alpha2 );
    true
    gap> alpha2;
    <A homomorphism of left modules>
    gap> Display( alpha2 );
       1

    the map is currently represented by the above 1 x 1 matrix
    gap> M_ := Kernel( alpha2 );
    <A cyclic left module presented by yet unknown relations for a cyclic generato\
    r>
    gap> alpha1 := KernelEmb( alpha2 );
    <A monomorphism of left modules>
    gap> seq := HomalgComplex( alpha2 );
    <An acyclic complex containing a single morphism of left modules at degrees
    [ 0 .. 1 ]>
    gap> Add( seq, alpha1 );
    gap> seq;
    <A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]>
    gap> IsShortExactSequence( seq );
    true
    gap> seq;
    <A short exact sequence containing 2 morphisms of left modules at degrees
    [ 0 .. 2 ]>
    gap> Display( seq );
    -------------------------
    at homology degree: 2
    Z/256Z/< ZmodnZObj(4,256) >
    -------------------------
       8

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 1
    Z/256Z/< ZmodnZObj(32,256) >
    -------------------------
       1

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 0
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
    gap> K := LeftPresentation( [ 2^7 ], R );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> L := RightPresentation( [ 2^4 ], R );
    <A cyclic right module on a cyclic generator satisfying 1 relation>
    gap> triangle := LHomHom( 4, seq, K, L, "t" );
    <An exact triangle containing 3 morphisms of left complexes at degrees
    [ 1, 2, 3, 1 ]>
    gap> lehs := LongSequence( triangle );
    <A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]>
    gap> ByASmallerPresentation( lehs );
    <A non-zero sequence containing 14 morphisms of left modules at degrees
    [ 0 .. 14 ]>
    gap> IsExactSequence( lehs );
    false
    gap> lehs;
    <A non-zero left acyclic complex containing
    14 morphisms of left modules at degrees [ 0 .. 14 ]>
    gap> Assert( 0, IsLeftAcyclic( lehs ) );
    gap> Display( lehs );
    -------------------------
    at homology degree: 14
    Z/256Z/< ZmodnZObj(4,256) >
    -------------------------
       4

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 13
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 12
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 11
    Z/256Z/< ZmodnZObj(4,256) >
    -------------------------
       4

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 10
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 9
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 8
    Z/256Z/< ZmodnZObj(4,256) >
    -------------------------
       4

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 7
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 6
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 5
    Z/256Z/< ZmodnZObj(4,256) >
    -------------------------
       4

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 4
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 3
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------
       2

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 2
    Z/256Z/< ZmodnZObj(4,256) >
    -------------------------
       8

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 1
    Z/256Z/< ZmodnZObj(16,256) >
    -------------------------
       1

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 0
    Z/256Z/< ZmodnZObj(8,256) >
    -------------------------

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