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### 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_⟨ 2⟩ (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> ZZ := HomalgRingOfIntegers(  );
Z
gap> R := LocalizeAt( ZZ , [ 2 ] );
Z_< 2 >
gap> Display( R );
<A local ring>
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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