This chapter should give you a quick guide to create your first example in **homalg**.

As the reader might notice, all examples in this manual will be either over ℤ or over one of its residue class rings \(ℤ/mℤ\). There are two reasons for this. The first reason is that **GAP** does not natively support rings other than ℤ in a *sufficient* way (--> Rings supported in a sufficient way).

The second and more important reason is to underline the fact the all effective homological constructions that are relevant for **Modules** have only as much to do with the Gröbnerbasis algorithm as they do with the Hermite algorithm for the ring ℤ; both algorithms are used to effectively solve inhomogeneous linear systems over the respective ring. And **Modules** is designed to use rings and matrices over these rings together with all their operations as a black box. In other words: Because **Modules** works for \(ℤ\), it works by its design for all other computable rings.

`gap> ExamplesForHomalg();`

To quickly create a ring for use with **Modules** enter

`ExamplesForHomalg();`

which will load the package **ExamplesForHomalg** (if installed) and provide a step by step guide to create the ring. For the full core functionality you need to install the packages **homalg**, **HomalgToCAS**, **IO_ForHomalg**, **RingsForHomalg**, **Gauss**, and **GaussForHomalg**. They are part of the **homalg** project.

The following example is taken from Section 2 of [BR06].

The computation takes place over the residue class ring \(R=ℤ/2^8ℤ\) using the generic support for residue class rings provided by the subpackage **ResidueClassRingForHomalg** of the **MatricesForHomalg** package. For a native support of the rings \(R=ℤ/p^nℤ\) use the **GaussForHomalg** package.

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> zz := HomalgRingOfIntegers( ); Z gap> Display( zz ); <An internal ring> gap> R := zz / 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 := LeftPresentation( [ 2^3 ], R ); <A cyclic left module presented by 1 relation for a cyclic generator> gap> Display( _M ); Z/( 256 )/< |[ 8 ]| > 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 ]| > -------------------------

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