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<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>
<div class="ChapSects"><a href="chap3.html#X7A489A5D79DA9E5C">3 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7AD67ACA80E44216">3.1 <span class="Heading">Spectral Filtrations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BB9DE017ECE6E86">3.1-1 <span class="Heading">ExtExt</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7EE63228803A04F1">3.1-2 <span class="Heading">Purity</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7816D6ED815ED641">3.1-3 <span class="Heading">A3_Purity</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X812EF8147AE16E72">3.1-4 <span class="Heading">TorExt-Grothendieck</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X784BC2567875830B">3.1-5 <span class="Heading">TorExt</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8021C33D85444081">3.1-6 <span class="Heading">CodegreeOfPurity</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X791E21F47805048A">3.1-7 <span class="Heading">HomHom</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X85CF19B87D1C375F">3.2 <span class="Heading">Commutative Algebra</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X781B1C0C80529B09">3.2-1 <span class="Heading">Eliminate</span></a>
</span>
</div></div>
</div>
<h3>3 <span class="Heading">Examples</span></h3>
<p><a id="X7AD67ACA80E44216" name="X7AD67ACA80E44216"></a></p>
<h4>3.1 <span class="Heading">Spectral Filtrations</span></h4>
<p><a id="X7BB9DE017ECE6E86" name="X7BB9DE017ECE6E86"></a></p>
<h5>3.1-1 <span class="Heading">ExtExt</span></h5>
<p>This is Example B.2 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x*y, y*z, z, 0, 0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^3*z,x^2*z^2,0, x*z^2, -z^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^4, x^3*z, 0, x^2*z, -x*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x*y, -y^2, x^2-1,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*z, -x*y*z, y*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 6, 5, Qxyz );</span>
<A 6 x 5 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := LeftPresentation( wmat );</span>
<A left module presented by 6 relations for 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y := Hom( Qxyz, W );</span>
<A right module on 5 generators satisfying yet unknown relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoWarning, 0 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, Y, "TensorY" );</span>
<The functor TensorY for f.p. modules and their maps over computable rings>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoWarning, 1 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := LeftDualizingFunctor( Qxyz );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">II_E := GrothendieckSpectralSequence( F, G, W );</span>
<A stable homological spectral sequence with sheets at levels
[ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
[ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_E );</span>
The associated transposed spectral sequence:
a homological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -3 .. 0 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
. . . .
. . . .
. . . .
---------
Level 2:
s s s s
. . . .
. . . .
. . . .
Now the spectral sequence of the bicomplex:
a homological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
* * * *
. * * *
. . . *
---------
Level 2:
* * s s
* * * *
. * * *
. . . *
---------
Level 3:
* s s s
* s s s
. . s *
. . . *
---------
Level 4:
s s s s
. s s s
. . s s
. . . s
<span class="GAPprompt">gap></span> <span class="GAPinput">filt := FiltrationBySpectralSequence( II_E, 0 );</span>
<An ascending filtration with degrees [ -3 .. 0 ] and graded parts:
0: <A non-zero left module presented by yet unknown relations for 23 generator\
s>
-1: <A non-zero left module presented by 37 relations for 22 generators>
-2: <A non-zero left module presented by 31 relations for 10 generators>
-3: <A non-zero left module presented by 33 relations for 5 generators>
of
<A non-zero left module presented by 102 relations for 37 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( filt );</span>
<An ascending filtration with degrees [ -3 .. 0 ] and graded parts:
0: <A non-zero left module presented by 26 relations for 16 generators>
-1: <A non-zero left module presented by 30 relations for 14 generators>
-2: <A non-zero left module presented by 18 relations for 7 generators>
-3: <A non-zero left module presented by 12 relations for 4 generators>
of
<A non-zero left module presented by 48 relations for 20 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := IsomorphismOfFiltration( filt );</span>
<A non-zero isomorphism of left modules>
</pre></div>
<p><a id="X7EE63228803A04F1" name="X7EE63228803A04F1"></a></p>
<h5>3.1-2 <span class="Heading">Purity</span></h5>
<p>This is Example B.3 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x*y, y*z, z, 0, 0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^3*z,x^2*z^2,0, x*z^2, -z^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^4, x^3*z, 0, x^2*z, -x*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x*y, -y^2, x^2-1,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*z, -x*y*z, y*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 6, 5, Qxyz );</span>
<A 6 x 5 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := LeftPresentation( wmat );</span>
<A left module presented by 6 relations for 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt := PurityFiltration( W );</span>
<The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:
0: <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\
generators>
-1: <A codegree-1-pure grade 1 left module presented by 4 relations for 3 gene\
rators>
-2: <A cyclic reflexively pure grade 2 left module presented by 2 relations fo\
r a cyclic generator>
-3: <A cyclic reflexively pure grade 3 left module presented by 3 relations fo\
r a cyclic generator>
of
<A non-pure rank 2 left module presented by 6 relations for 5 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">W;</span>
<A non-pure rank 2 left module presented by 6 relations for 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">II_E := SpectralSequence( filt );</span>
<A stable homological spectral sequence with sheets at levels
[ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
[ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_E );</span>
The associated transposed spectral sequence:
a homological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -3 .. 0 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
. . . .
. . . .
. . . .
---------
Level 2:
s . . .
. . . .
. . . .
. . . .
Now the spectral sequence of the bicomplex:
a homological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
* * * *
. * * *
. . . *
---------
Level 2:
s . . .
* s . .
. * * .
. . . *
---------
Level 3:
s . . .
* s . .
. . s .
. . . *
---------
Level 4:
s . . .
. s . .
. . s .
. . . s
<span class="GAPprompt">gap></span> <span class="GAPinput">m := IsomorphismOfFiltration( filt );</span>
<A non-zero isomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( Range( m ), W );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Source( m );</span>
<A left module presented by 12 relations for 9 generators (locked)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( last );</span>
0, 0,x, -y,0,1, 0, 0, 0,
x*y,0,-z,0, 0,0, 0, 0, 0,
x^2,0,0, -z,1,0, 0, 0, 0,
0, 0,0, 0, y,-z,0, 0, 0,
0, 0,0, 0, 0,x, -y, -1, 0,
0, 0,0, 0, x,0, -z, 0, -1,
0, 0,0, 0, 0,-y,x^2-1,0, 0,
0, 0,0, 0, 0,0, 0, z, 0,
0, 0,0, 0, 0,0, 0, y-1,0,
0, 0,0, 0, 0,0, 0, 0, z,
0, 0,0, 0, 0,0, 0, 0, y,
0, 0,0, 0, 0,0, 0, 0, x
Cokernel of the map
Q[x,y,z]^(1x12) --> Q[x,y,z]^(1x9),
currently represented by the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( filt );</span>
Degree 0:
0, 0,x, -y,
x*y,0,-z,0,
x^2,0,0, -z
Cokernel of the map
Q[x,y,z]^(1x3) --> Q[x,y,z]^(1x4),
currently represented by the above matrix
----------
Degree -1:
y,-z,0,
0,x, -y,
x,0, -z,
0,-y,x^2-1
Cokernel of the map
Q[x,y,z]^(1x4) --> Q[x,y,z]^(1x3),
currently represented by the above matrix
----------
Degree -2:
Q[x,y,z]/< z, y-1 >
----------
Degree -3:
Q[x,y,z]/< z, y, x >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
1, 0, 0, 0, 0,
0, 1, 0, 0, 0,
0, -y, -1, 0, 0,
0, -x, 0, -1, 0,
-x^2,-x*z, 0, -z, 0,
0, 0, x, -y, 0,
0, 0, 0, 0, -1,
0, 0, x^2,-x*y,y,
-x^3,-x^2*z,0, -x*z,z
the map is currently represented by the above 9 x 5 matrix
</pre></div>
<p><a id="X7816D6ED815ED641" name="X7816D6ED815ED641"></a></p>
<h5>3.1-3 <span class="Heading">A3_Purity</span></h5>
<p>This is Example B.4 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );</span>
Q[x,y,z]<Dx,Dy,Dz>
<span class="GAPprompt">gap></span> <span class="GAPinput">nmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">3*Dy*Dz-Dz^2+Dx+3*Dy-Dz, 3*Dy*Dz-Dz^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">Dx*Dz+Dz^2+Dz, Dx*Dz+Dz^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">Dx*Dy, 0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">Dz^2-Dx+Dz, 3*Dx*Dy+Dz^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">Dx^2, 0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">-Dz^2+Dx-Dz, 3*Dx^2-Dz^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">Dz^3-Dx*Dz+Dz^2, Dz^3, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">2*x*Dz^2-2*x*Dx+2*x*Dz+3*Dx+3*Dz+3,2*x*Dz^2+3*Dx+3*Dz\</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 8, 2, A3 );</span>
<A 8 x 2 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := LeftPresentation( nmat );</span>
<A left module presented by 8 relations for 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt := PurityFiltration( N );</span>
<The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:
0: <A zero left module>
-1: <A cyclic reflexively pure grade 1 left module presented by 1 relation for\
a cyclic generator>
-2: <A cyclic reflexively pure grade 2 left module presented by 2 relations fo\
r a cyclic generator>
-3: <A cyclic reflexively pure grade 3 left module presented by 3 relations fo\
r a cyclic generator>
of
<A non-pure grade 1 left module presented by 8 relations for 2 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">II_E := SpectralSequence( filt );</span>
<A stable homological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -4 .. 0 ]x
[ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_E );</span>
The associated transposed spectral sequence:
a homological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -4 .. 0 ] ]
---------
Level 0:
* * * *
. * * *
. . * *
. . . *
. . . *
---------
Level 1:
* * * *
. . . .
. . . .
. . . .
. . . .
---------
Level 2:
s . . .
. . . .
. . . .
. . . .
. . . .
Now the spectral sequence of the bicomplex:
a homological spectral sequence at bidegrees
[ [ -4 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:
* * * * *
. . * * *
. . . * *
. . . . *
---------
Level 1:
* * * * *
. . * * *
. . . * *
. . . . .
---------
Level 2:
. s . . .
. . s . .
. . . s .
. . . . .
<span class="GAPprompt">gap></span> <span class="GAPinput">m := IsomorphismOfFiltration( filt );</span>
<A non-zero isomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( Range( m ), N );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Source( m );</span>
<A left module presented by 6 relations for 3 generators (locked)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( last );</span>
Dx,1/3,1/216*x,
0, Dy, -1/144,
0, Dx, 1/48,
0, 0, Dz,
0, 0, Dy,
0, 0, Dx
Cokernel of the map
R^(1x6) --> R^(1x3), ( for R := Q[x,y,z]<Dx,Dy,Dz> )
currently represented by the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( filt );</span>
Degree 0:
0
----------
Degree -1:
Q[x,y,z]<Dx,Dy,Dz>/< Dx >
----------
Degree -2:
Q[x,y,z]<Dx,Dy,Dz>/< Dy, Dx >
----------
Degree -3:
Q[x,y,z]<Dx,Dy,Dz>/< Dz, Dy, Dx >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
1, 1,
3*Dz+3, 3*Dz,
144*Dz^2-144*Dx+144*Dz,144*Dz^2
the map is currently represented by the above 3 x 2 matrix
</pre></div>
<p><a id="X812EF8147AE16E72" name="X812EF8147AE16E72"></a></p>
<h5>3.1-4 <span class="Heading">TorExt-Grothendieck</span></h5>
<p>This is Example B.5 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x*y, y*z, z, 0, 0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^3*z,x^2*z^2,0, x*z^2, -z^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^4, x^3*z, 0, x^2*z, -x*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x*y, -y^2, x^2-1,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*z, -x*y*z, y*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 6, 5, Qxyz );</span>
<A 6 x 5 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := LeftPresentation( wmat );</span>
<A left module presented by 6 relations for 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, W, "TensorW" );</span>
<The functor TensorW for f.p. modules and their maps over computable rings>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := LeftDualizingFunctor( Qxyz );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">II_E := GrothendieckSpectralSequence( F, G, W );</span>
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
[ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_E );</span>
The associated transposed spectral sequence:
a cohomological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -3 .. 0 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
. . . .
. . . .
. . . .
---------
Level 2:
s s s s
. . . .
. . . .
. . . .
Now the spectral sequence of the bicomplex:
a cohomological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
* * * *
. * * *
. . . *
---------
Level 2:
* * s s
* * * *
. * * *
. . . *
---------
Level 3:
* s s s
. s s s
. . s *
. . . s
---------
Level 4:
s s s s
. s s s
. . s s
. . . s
<span class="GAPprompt">gap></span> <span class="GAPinput">filt := FiltrationBySpectralSequence( II_E, 0 );</span>
<A descending filtration with degrees [ -3 .. 0 ] and graded parts:
-3: <A non-zero cyclic torsion left module presented by yet unknown relations \
for a cyclic generator>
-2: <A non-zero left module presented by 17 relations for 6 generators>
-1: <A non-zero left module presented by 27 relations for 12 generators>
0: <A non-zero left module presented by 13 relations for 10 generators>
of
<A left module presented by yet unknown relations for 49 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( filt );</span>
<A descending filtration with degrees [ -3 .. 0 ] and graded parts:
-3: <A non-zero cyclic torsion left module presented by 3 relations for a cycl\
ic generator>
-2: <A non-zero left module presented by 12 relations for 4 generators>
-1: <A non-zero left module presented by 21 relations for 8 generators>
0: <A non-zero left module presented by 11 relations for 10 generators>
of
<A non-zero left module presented by 27 relations for 14 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := IsomorphismOfFiltration( filt );</span>
<A non-zero isomorphism of left modules>
</pre></div>
<p><a id="X784BC2567875830B" name="X784BC2567875830B"></a></p>
<h5>3.1-5 <span class="Heading">TorExt</span></h5>
<p>This is Example B.6 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x*y, y*z, z, 0, 0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^3*z,x^2*z^2,0, x*z^2, -z^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^4, x^3*z, 0, x^2*z, -x*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x*y, -y^2, x^2-1,\</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*z, -x*y*z, y*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 6, 5, Qxyz );</span>
<A 6 x 5 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := LeftPresentation( wmat );</span>
<A left module presented by 6 relations for 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := Resolution( W );</span>
<A right acyclic complex containing 3 morphisms of left modules at degrees
[ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">GP := Hom( P );</span>
<A cocomplex containing 3 morphisms of right modules at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FGP := GP * P;</span>
<A cocomplex containing 3 morphisms of left complexes at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">BC := HomalgBicomplex( FGP );</span>
<A bicocomplex containing left modules at bidegrees [ 0 .. 3 ]x[ -3 .. 0 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">p_degrees := ObjectDegreesOfBicomplex( BC )[1];</span>
[ 0 .. 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );</span>
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
[ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_E );</span>
The associated transposed spectral sequence:
a cohomological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -3 .. 0 ] ]
---------
Level 0:
* * * *
* * * *
* * * *
* * * *
---------
Level 1:
* * * *
. . . .
. . . .
. . . .
---------
Level 2:
s s s s
. . . .
. . . .
. . . .
Now the spectral sequence of the bicomplex:
a cohomological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:
* * * *
* * * *
* * * *
* * * *
---------
Level 1:
* * * *
* * * *
* * * *
* * * *
---------
Level 2:
* * s s
* * * *
. * * *
. . . *
---------
Level 3:
* s s s
. s s s
. . s *
. . . s
---------
Level 4:
s s s s
. s s s
. . s s
. . . s
<span class="GAPprompt">gap></span> <span class="GAPinput">filt := FiltrationBySpectralSequence( II_E, 0 );</span>
<A descending filtration with degrees [ -3 .. 0 ] and graded parts:
-3: <A non-zero cyclic torsion left module presented by yet unknown relations \
for a cyclic generator>
-2: <A non-zero left module presented by 15 relations for 6 generators>
-1: <A non-zero left module presented by 27 relations for 13 generators>
0: <A non-zero left module presented by 13 relations for 10 generators>
of
<A left module presented by yet unknown relations for 31 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( filt );</span>
<A descending filtration with degrees [ -3 .. 0 ] and graded parts:
-3: <A non-zero cyclic torsion left module presented by 3 relations for a cycl\
ic generator>
-2: <A non-zero left module presented by 11 relations for 4 generators>
-1: <A non-zero left module presented by 23 relations for 9 generators>
0: <A non-zero left module presented by 11 relations for 10 generators>
of
<A non-zero left module presented by 24 relations for 12 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := IsomorphismOfFiltration( filt );</span>
<A non-zero isomorphism of left modules>
</pre></div>
<p><a id="X8021C33D85444081" name="X8021C33D85444081"></a></p>
<h5>3.1-6 <span class="Heading">CodegreeOfPurity</span></h5>
<p>This is Example B.7 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";</span>
Q[x,y,z]
<span class="GAPprompt">gap></span> <span class="GAPinput">vmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x,-z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x*z,z^2,y,0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^2,x*z,0,y \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 3, 4, Qxyz );</span>
<A 3 x 4 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">V := LeftPresentation( vmat );</span>
<A non-torsion left module presented by 3 relations for 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 0, x,-y, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x*y,y*z,z,0, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^2,x*z,0,z \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 3, 4, Qxyz );</span>
<A 3 x 4 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := LeftPresentation( wmat );</span>
<A non-torsion left module presented by 3 relations for 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Rank( V );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Rank( W );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension( V );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension( W );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTorsionFreeness( V );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfTorsionFreeness( W );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">CodegreeOfPurity( V );</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CodegreeOfPurity( W );</span>
[ 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">filtV := PurityFiltration( V );</span>
<The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts:
0: <A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 ge\
nerators>
-1: <A zero left module>
-2: <A zero left module>
of
<A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 gener\
ators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">filtW := PurityFiltration( W );</span>
<The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts:
0: <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\
generators>
-1: <A zero left module>
-2: <A zero left module>
of
<A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4 ge\
nerators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">II_EV := SpectralSequence( filtV );</span>
<A stable homological spectral sequence with sheets at levels
[ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
[ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_EV );</span>
The associated transposed spectral sequence:
a homological spectral sequence at bidegrees
[ [ 0 .. 2 ], [ -3 .. 0 ] ]
---------
Level 0:
* * *
* * *
* * *
. * *
---------
Level 1:
* * *
. . .
. . .
. . .
---------
Level 2:
s . .
. . .
. . .
. . .
Now the spectral sequence of the bicomplex:
a homological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 2 ] ]
---------
Level 0:
* * * *
* * * *
. * * *
---------
Level 1:
* * * *
* * * *
. . * *
---------
Level 2:
* . . .
* . . .
. . * *
---------
Level 3:
* . . .
. . . .
. . . *
---------
Level 4:
. . . .
. . . .
. . . s
<span class="GAPprompt">gap></span> <span class="GAPinput">II_EW := SpectralSequence( filtW );</span>
<A stable homological spectral sequence with sheets at levels
[ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
[ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_EW ); </span>
The associated transposed spectral sequence:
a homological spectral sequence at bidegrees
[ [ 0 .. 2 ], [ -3 .. 0 ] ]
---------
Level 0:
* * *
* * *
. * *
. . *
---------
Level 1:
* * *
. . .
. . .
. . .
---------
Level 2:
s . .
. . .
. . .
. . .
Now the spectral sequence of the bicomplex:
a homological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 2 ] ]
---------
Level 0:
* * * *
. * * *
. . * *
---------
Level 1:
* * * *
. * * *
. . . *
---------
Level 2:
* . . .
. * . .
. . . *
---------
Level 3:
* . . .
. . . .
. . . *
---------
Level 4:
. . . .
. . . .
. . . s
</pre></div>
<p><a id="X791E21F47805048A" name="X791E21F47805048A"></a></p>
<h5>3.1-7 <span class="Heading">HomHom</span></h5>
<p>This corresponds to the example of Section 2 in <a href="chapBib.html#biBBREACA">[BR06]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgRingOfIntegersInExternalGAP( ) / 2^8;</span>
Z/( 256 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( R );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( [ 2^5 ], R );</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/( 256 )/< |[ 32 ]| >
<span class="GAPprompt">gap></span> <span class="GAPinput">M;</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">_M := LeftPresentation( [ 2^3 ], R );</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/( 256 )/< |[ 8 ]| >
<span class="GAPprompt">gap></span> <span class="GAPinput">_M;</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha2 := HomalgMap( [ 1 ], M, _M );</span>
<A "homomorphism" of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( alpha2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha2;</span>
<A homomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( alpha2 );</span>
[ [ 1 ] ]
modulo [ 256 ]
the map is currently represented by the above 1 x 1 matrix
<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">seq := HomalgComplex( alpha2 );</span>
<An acyclic complex containing a single morphism of left modules at degrees
[ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Add( seq, alpha1 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">seq;</span>
<A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsShortExactSequence( seq );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">seq;</span>
<A short exact sequence containing 2 morphisms of left modules at degrees
[ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( seq );</span>
-------------------------
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 )/< |[ 32 ]| >
-------------------------
[ [ 1 ] ]
modulo [ 256 ]
the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 0
Z/( 256 )/< |[ 8 ]| >
-------------------------
<span class="GAPprompt">gap></span> <span class="GAPinput">K := LeftPresentation( [ 2^7 ], R );</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := RightPresentation( [ 2^4 ], R );</span>
<A cyclic right module on a cyclic generator satisfying 1 relation>
<span class="GAPprompt">gap></span> <span class="GAPinput">triangle := LHomHom( 4, seq, K, L, "t" );</span>
<An exact triangle containing 3 morphisms of left complexes at degrees
[ 1, 2, 3, 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">lehs := LongSequence( triangle );</span>
<A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( lehs );</span>
<A non-zero sequence containing 14 morphisms of left modules at degrees
[ 0 .. 14 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsExactSequence( lehs );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">lehs;</span>
<A non-zero left acyclic complex containing
14 morphisms of left modules at degrees [ 0 .. 14 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Assert( 0, IsLeftAcyclic( lehs ) );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( lehs );</span>
-------------------------
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 ]| >
-------------------------
</pre></div>
<p><a id="X85CF19B87D1C375F" name="X85CF19B87D1C375F"></a></p>
<h4>3.2 <span class="Heading">Commutative Algebra</span></h4>
<p><a id="X781B1C0C80529B09" name="X781B1C0C80529B09"></a></p>
<h5>3.2-1 <span class="Heading">Eliminate</span></h5>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,l,m";</span>
Q[x,y,z,l,m]
<span class="GAPprompt">gap></span> <span class="GAPinput">var := Indeterminates( R );</span>
[ x, y, z, l, m ]
<span class="GAPprompt">gap></span> <span class="GAPinput">x := var[1];; y := var[2];; z := var[3];; l := var[4];; m := var[5];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ];</span>
[ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ]
<span class="GAPprompt">gap></span> <span class="GAPinput">e := Eliminate( L, [ l, m ] );</span>
<A non-zero right regular 3 x 1 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( e );</span>
4*y+z,
4*x-5*z,
21*z^2-8
<span class="GAPprompt">gap></span> <span class="GAPinput">I := LeftSubmodule( e );</span>
<A torsion-free (left) ideal given by 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I );</span>
4*y+z,
4*x-5*z,
21*z^2-8
A (left) ideal generated by the 3 entries of the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">J := LeftSubmodule( "x+y-z, -2*z-3*y+x, x^2+y^2+z^2-1", R );</span>
<A torsion-free (left) ideal given by 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">I = J;</span>
true
</pre></div>
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