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<title>GAP (LocalizeRingForHomalg) - Chapter 3: Quick Start</title>
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<div class="ChapSects"><a href="chap3_mj.html#X7EB860EC84DFC71E">3 <span class="Heading">Quick Start</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X799572BF810091BC">3.1 <span class="Heading">Localization of ℤ</span></a>
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<h3>3 <span class="Heading">Quick Start</span></h3>

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<h4>3.1 <span class="Heading">Localization of ℤ</span></h4>

<p>The following example is taken from Section 2 of <a href="chapBib_mj.html#biBBREACA">[BR06]</a>. <br /> <br /> The computation takes place over the local ring <span class="SimpleMath">\(R=ℤ_{\langle 2\rangle}\)</span> (i.e. ℤ localized at the maximal ideal generated by <span class="SimpleMath">\(2\)</span>).</p>

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

<p>We want to lead your attention to the commands <code class="keyw">LocalizeAt</code> and <code class="keyw">HomalgLocalMatrix</code>. 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 <strong class="pkg">homalg</strong>.</p>


<div class="example"><pre>
  
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "LocalizeRingForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers(  );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">R := LocalizeAt( zz , [ 2 ] );</span>
Z_< 2 >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( R );</span>
<A local ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "Modules" );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( HomalgMatrix( [ 2^5 ], R ) );</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">_M := LeftPresentation( HomalgMatrix( [ 2^3 ], R ) );</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha2 := HomalgMap( HomalgMatrix( [ 1 ], R ), M, _M );</span>
<A "homomorphism" of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">M_ := Kernel( alpha2 );</span>
<A cyclic left module presented by yet unknown relations for a cyclic generato\
r>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha1 := KernelEmb( alpha2 );</span>
<A monomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M_ );</span>
Z_< 2 >/< -4/1 >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( alpha1 );</span>
[ [  8 ] ]
/ 1

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