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<div class="ChapSects"><a href="chap12_mj.html#X7A489A5D79DA9E5C">12 Examples</a>
<div class="ContSect"> <a href="chap12_mj.html#X7BB9DE017ECE6E86">12.1 ExtExt</a>

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<div class="ContSect"> <a href="chap12_mj.html#X7EE63228803A04F1">12.2 Purity</a>

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<div class="ContSect"> <a href="chap12_mj.html#X812EF8147AE16E72">12.3 TorExt-Grothendieck</a>

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<div class="ContSect"> <a href="chap12_mj.html#X784BC2567875830B">12.4 TorExt</a>

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<h3>12 Examples</h3>

<a id="X7BB9DE017ECE6E86" name="X7BB9DE017ECE6E86"></a>

<h4>12.1 ExtExt</h4>

This corresponds to Example B.2 in <a href="chapBib_mj.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> N := Hom( zz, M );
<A rank 1 right module on 4 generators satisfying yet unknown relations>
gap> F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, N, "TensorN" );
<The functor TensorN for f.p. modules and their maps over computable rings>
gap> G := LeftDualizingFunctor( zz );;
gap> II_E := GrothendieckSpectralSequence( F, G, M );
<A stable homological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a homological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
. .
---------
Level 2:

s s
. .

Now the spectral sequence of the bicomplex:

a homological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
. s
---------
Level 2:

s s
. s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A non-torsion left module presented by 3 relations for 4 generators>
 -1: <A non-zero left module presented by 21 relations for 8 generators>
of
<A non-zero left module presented by 31 relations for 19 generators>>
gap> ByASmallerPresentation( filt );
<An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A rank 1 left module presented by 2 relations for 3 generators>
 -1: <A non-zero torsion left module presented by 6 relations for 6 generators>
of
<A rank 1 left module presented by 8 relations for 9 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

<a id="X7EE63228803A04F1" name="X7EE63228803A04F1"></a>

<h4>12.2 Purity</h4>

This corresponds to Example B.3 in <a href="chapBib_mj.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> filt := PurityFiltration( M );
<The ascending purity filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A free left module of rank 1 on a free generator>
 -1: <A non-zero torsion left module presented by 2 relations for 2 generators>
of
<A non-pure rank 1 left module presented by 2 relations for 3 generators>>
gap> M;
<A non-pure rank 1 left module presented by 2 relations for 3 generators>
gap> II_E := SpectralSequence( filt );
<A stable homological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a homological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
. .
---------
Level 2:

s .
. .

Now the spectral sequence of the bicomplex:

a homological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
. s
---------
Level 2:

s .
. s
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
gap> IsIdenticalObj( Range( m ), M );
true
gap> Source( m );
<A non-torsion left module presented by 2 relations for 3 generators (locked)>
gap> Display( last );
[ [ 0, 2, 0 ],
 [ 0, 0, 12 ] ]

Cokernel of the map

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

currently represented by the above matrix
gap> Display( filt );
Degree 0:

Z^(1 x 1)
----------
Degree -1:

Z/< 2 > + Z/< 12 >
</pre></div>

<a id="X812EF8147AE16E72" name="X812EF8147AE16E72"></a>

<h4>12.3 TorExt-Grothendieck</h4>

This corresponds to Example B.5 in <a href="chapBib_mj.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, M, "TensorM" );
<The functor TensorM for f.p. modules and their maps over computable rings>
gap> G := LeftDualizingFunctor( zz );;
gap> II_E := GrothendieckSpectralSequence( F, G, M );
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a cohomological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
. .
---------
Level 2:

s s
. .

Now the spectral sequence of the bicomplex:

a cohomological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
. s
---------
Level 2:

s s
. s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:

-1: <A non-zero left module presented by yet unknown relations for 6 generator\
s>

0: <A non-zero left module presented by yet unknown relations for 4 generators\
>
of
<A left module presented by yet unknown relations for 14 generators>>
gap> ByASmallerPresentation( filt );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:
 -1: <A non-zero torsion left module presented by 4 relations
 for 4 generators>
 0: <A rank 1 left module presented by 2 relations for 3 generators>
of
<A rank 1 left module presented by 6 relations for 7 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

<a id="X784BC2567875830B" name="X784BC2567875830B"></a>

<h4>12.4 TorExt</h4>

This corresponds to Example B.6 in <a href="chapBib_mj.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> P := Resolution( M );
<A non-zero right acyclic complex containing a single morphism of left modules\
at degrees [ 0 .. 1 ]>
gap> GP := Hom( P );
<A non-zero acyclic cocomplex containing a single morphism of right modules at\
degrees [ 0 .. 1 ]>
gap> FGP := GP * P;
<A non-zero acyclic cocomplex containing a single morphism of left complexes a\
t degrees [ 0 .. 1 ]>
gap> BC := HomalgBicomplex( FGP );
<A non-zero bicocomplex containing left modules at bidegrees [ 0 .. 1 ]x
[ -1 .. 0 ]>
gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1];
[ 0, 1 ]
gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a cohomological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
. .
---------
Level 2:

s s
. .

Now the spectral sequence of the bicomplex:

a cohomological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

* *
* *
---------
Level 1:

* *
* *
---------
Level 2:

s s
. s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:

-1: <A non-zero torsion left module presented by yet unknown relations for
 4 generators>
0: <A rank 1 left module presented by 3 relations for 4 generators>
of
<A left module presented by yet unknown relations for 13 generators>>
gap> ByASmallerPresentation( filt );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:
 -1: <A non-zero torsion left module presented by 4 relations
 for 4 generators>
 0: <A rank 1 left module presented by 2 relations for 3 generators>
of
<A rank 1 left module presented by 6 relations for 7 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

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