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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