chap3.txt

  
  3 Quick Start
  
  This  chapter  should give you a quick guide to create your first example in
  homalg.
  
  
  3.1 Why are all examples in this manual over ℤ or ℤ/mℤ?
  
  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 (--> 1.1-2).
  
  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.
  
  
  3.2 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.
  
  
  3.3 A typical example
  
  
  3.3-1 HomHom
  
  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.
  
    Example  
    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 ]| > 
    -------------------------
    [ [  24 ] ]
    
    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 ]| >
    -------------------------