## <#GAPDoc Label="HomHom">
## <Subsection Label="HomHom">
## <Heading>HomHom</Heading>
## This corresponds to the example of Section 2 in <Cite Key="BREACA"/>.
## <Example><![CDATA[
## gap> R := HomalgRingOfIntegersInExternalGAP( ) / 2^8;
## Z/( 256 )
## gap> Display( R );
## <A residue class ring>
## gap> M := LeftPresentation( [ 2^5 ], R );
## <A cyclic left module presented by 1 relation for a cyclic generator>
## gap> Display( M );
## Z/( 256 )/< |[ 32 ]| >
## 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/( 256 )/< |[ 8 ]| >
## 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 ] ]
##
## modulo [ 256 ]
##
## 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/( 256 )/< |[ 4 ]| >
## -------------------------
## [ [ 8 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 1
## Z/( 256 )/< |[ 32 ]| >
## -------------------------
## [ [ 1 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 0
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## 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/( 256 )/< |[ 4 ]| >
## -------------------------
## [ [ 4 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 13
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 12
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 11
## Z/( 256 )/< |[ 4 ]| >
## -------------------------
## [ [ 4 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 10
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 9
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 8
## Z/( 256 )/< |[ 4 ]| >
## -------------------------
## [ [ 4 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 7
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 6
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 5
## Z/( 256 )/< |[ 4 ]| >
## -------------------------
## [ [ 4 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 4
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 3
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## [ [ 2 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 2
## Z/( 256 )/< |[ 4 ]| >
## -------------------------
## [ [ 8 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 1
## Z/( 256 )/< |[ 16 ]| >
## -------------------------
## [ [ 1 ] ]
##
## modulo [ 256 ]
##
## the map is currently represented by the above 1 x 1 matrix
## ------------v------------
## at homology degree: 0
## Z/( 256 )/< |[ 8 ]| >
## -------------------------
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
LoadPackage( "RingsForHomalg" );
R := HomalgRingOfIntegersInDefaultCAS( 2^8 );
LoadPackage( "Modules" );
M := LeftPresentation( [ 2^5 ], R );
_M := LeftPresentation( [ 2^3 ], R );
alpha2 := HomalgMap( [ 1 ], M, _M );
M_ := Kernel( alpha2 );
alpha1 := KernelEmb( alpha2 );
seq := HomalgComplex( alpha2 );
Add( seq, alpha1 );
IsShortExactSequence( seq );
K := LeftPresentation( [ 2^7 ], R );
L := RightPresentation( [ 2^4 ], R );
triangle := LHomHom( 4, seq, K, L, "t" );
lehs := LongSequence( triangle );
ByASmallerPresentation( lehs );
IsExactSequence( lehs );
Assert( 0, IsLeftAcyclic( lehs ) );
