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### 12 Examples

#### 12.1 ExtExt

This corresponds to Example B.2 in [Bar].

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 33 relations for 8 generators>
of
<A non-zero left module presented by 27 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>


#### 12.2 Purity

This corresponds to Example B.3 in [Bar].

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 >


#### 12.3 TorExt-Grothendieck

This corresponds to Example B.5 in [Bar].

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 9 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 29 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>


#### 12.4 TorExt

This corresponds to Example B.6 in [Bar].

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
10 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>

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