GAP (LocalizeRingForHomalg) - Chapter 3: Quick Start

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3 Quick Start
3.1 Localization of ℤ

### 3 Quick Start

#### 3.1 Localization of ℤ

The following example is taken from Section 2 of [BR06].
The computation takes place over the local ring \(R=ℤ_{\langle 2\rangle}\) (i.e. ℤ localized at the maximal ideal generated by \(2\)).

Here we compute the (infinite) long exact homology sequence of the covariant functor \(Hom(Hom(-,R/2^7R),R/2^4R)\) (and its left derived functors) applied to the short exact sequence
\(0 -> M_=R/2^2R --alpha_1--> M=R/2^5R --alpha_2--> \_M=R/2^3R -> 0\).

We want to lead your attention to the commands `LocalizeAt` and `HomalgLocalMatrix`. The first one creates a localized ring from a global one and generators of a maximal ideal and the second one creates a local matrix from a global matrix. The other commands used here are well known from homalg.

    gap> LoadPackage( "LocalizeRingForHomalg" );;
    gap> ZZ := HomalgRingOfIntegers(  );
    Z
    gap> R := LocalizeAt( ZZ , [ 2 ] );
    Z_< 2 >
    gap> Display( R );
    <A local ring>
    gap> LoadPackage( "Modules" );
    true
    gap> M := LeftPresentation( HomalgMatrix( [ 2^5 ], R ) );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> _M := LeftPresentation( HomalgMatrix( [ 2^3 ], R ) );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> alpha2 := HomalgMap( HomalgMatrix( [ 1 ], R ), M, _M );
    <A "homomorphism" of left modules>
    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> Display( M_ );
    Z_< 2 >/< -4/1 >
    gap> Display( alpha1 );
    [ [  24 ] ]
    / 1

    the map is currently represented by the above 1 x 1 matrix
    gap> ByASmallerPresentation( M_ );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> Display( M_ );
    Z_< 2 >/< 4/1 >

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