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products/Sources/formale Sprachen/GAP/pkg/homalg/doc/ (GAP Algebra Version 4.15.1^©) Datei vom 10.0.2024 mit Größe 35 kB

Quelle chap6_mj.html

Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/homalg/doc/chap6_mj.html

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<p>In the

<div class="example"><pre>
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[ [ bythe matrix
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Z^(1 x 2)
-------------------------
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    0  0

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------------------
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-----
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true
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<h5>6

<class=func> class"width100"<body>
<p>Returns: a <strong class="pkg">homalg</strong> complex</p>

<p>It invokes <code class="code">ByASmallerPresentation</code> for <strong class="pkg">homalg</strong> (static) objects.</p>

<div class="example"><pre>
InstallMethod( ByASmallerPresentation,
        "for homalg complexes",
        [ IsHomalgComplex ],

  function( C )

    List( ObjectsOfComplex( C ), ByASmallerPresentation );

    if Length( ObjectDegreesOfComplex( C ) ) > 1 then
        List( MorphismsOfComplex( C ), DecideZero );
    fi;

    IsZero( C );

    return C;

end );
</pre></div>

<p>This method performs side effects on its argument <var class="Arg">C</var> and returns it.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, zz );</span>
<A 2 x 3 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( M );</span>
<A non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ 2, 3, 4, 5,   6, 7, 8, 9 ]", 2, 4, zz );</span>
<A 2 x 4 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := LeftPresentation( N );</span>
<A non-torsion left module presented by 2 relations for 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 2, 4, 6, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9  \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 3, 4, zz );</span>
<A 3 x 4 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := HomalgMap( mat, M, N );</span>
<A "homomorphism" of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( phi );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">phi;</span>
<A homomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := HomalgComplex( phi );</span>
<A non-zero acyclic complex containing a single morphism of left modules at de\
grees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span>
-------------------------
at homology degree: 1
[ [  2,  3,  4 ],
  [  5,  6,  7 ] ]

Cokernel of the map

Z^(1x2) --> Z^(1x3),

currently represented by the above matrix
-------------------------
[ [  0,  3,  6,  9 ],
  [  0,  2,  4,  6 ],
  [  0,  3,  6,  9 ] ]

the map is currently represented by the above 3 x 4 matrix
------------v------------
at homology degree: 0
[ [  2,  3,  4,  5 ],
  [  6,  7,  8,  9 ] ]

Cokernel of the map

Z^(1x2) --> Z^(1x4),

currently represented by the above matrix
-------------------------
</pre></div>

<p>And now:</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( C );</span>
<A non-zero acyclic complex containing a single morphism of left modules at de\
grees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span>
-------------------------
at homology degree: 1
Z/< 3 > + Z^(1 x 1)
-------------------------
[ [  0,  0,  0 ],
  [  2,  0,  0 ] ]

the map is currently represented by the above 2 x 3 matrix
------------v------------
at homology degree: 0
Z/< 4 > + Z^(1 x 2)
-------------------------
</pre></div>

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