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<a id="X78D1062D78BE08C1" name="X78D1062D78BE08C1"></a>
<div class="ChapSects"><a href="chap10.html#X78D1062D78BE08C1">10 Functors</a>
<div class="ContSect"> <a href="chap10.html#X7E41BC437F2B76E1">10.1 Functors: Category and Representations</a>

</div>
<div class="ContSect"> <a href="chap10.html#X86EE897086995E47">10.2 Functors: Constructors</a>

</div>
<div class="ContSect"> <a href="chap10.html#X7A21845C7C536717">10.3 Functors: Attributes</a>

</div>
<div class="ContSect"> <a href="chap10.html#X7D83D0EB87D2D872">10.4 Basic Functors</a>

<div class="ContSSBlock">
 <a href="chap10.html#X7B9FE8BF80D47B6E">10.4-1 functor_Cokernel</a>
 <a href="chap10.html#X875F177A82BF9B8B">10.4-2 Cokernel</a>
 <a href="chap10.html#X7A5B3B307B334706">10.4-3 functor_ImageObject</a>
 <a href="chap10.html#X7E3FF900821DCBE6">10.4-4 ImageObject</a>
 <a href="chap10.html#X85C128B37E76827F">10.4-5 Kernel</a>
 <a href="chap10.html#X7E6CDE7E85F09122">10.4-6 DefectOfExactness</a>
 <a href="chap10.html#X7B93718087EFD69B">10.4-7 Functor_Hom</a>
 <a href="chap10.html#X80015C78876B4F1E">10.4-8 Hom</a>
 <a href="chap10.html#X7A1A077D8268FADE">10.4-9 Functor_TensorProduct</a>
 <a href="chap10.html#X87EB0B4A852CF4C6">10.4-10 TensorProduct</a>
 <a href="chap10.html#X7D007A7079F7BEE3">10.4-11 Functor_Ext</a>
 <a href="chap10.html#X8692578881E71913">10.4-12 Ext</a>
 <a href="chap10.html#X821034FE80907E8D">10.4-13 Functor_Tor</a>
 <a href="chap10.html#X79821906875CF49E">10.4-14 Tor</a>
 <a href="chap10.html#X84C60D997A79524E">10.4-15 Functor_RHom</a>
 <a href="chap10.html#X7D8BDC0C817C10AB">10.4-16 RHom</a>
 <a href="chap10.html#X806251E3836C00B9">10.4-17 Functor_LTensorProduct</a>
 <a href="chap10.html#X7C12DA648798E77E">10.4-18 LTensorProduct</a>
 <a href="chap10.html#X7ACC6A7C86E4354C">10.4-19 Functor_HomHom</a>
 <a href="chap10.html#X84557A6B79382720">10.4-20 Functor_LHomHom</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X815BF6DA7FD5D44B">10.5 Tool Functors</a>

</div>
<div class="ContSect"> <a href="chap10.html#X879135AC8330C509">10.6 Other Functors</a>

</div>
<div class="ContSect"> <a href="chap10.html#X7DACD68E7E5FA324">10.7 Functors: Operations and Functions</a>

</div>
</div>

<h3>10 Functors</h3>

<a id="X7E41BC437F2B76E1" name="X7E41BC437F2B76E1"></a>

<h4>10.1 Functors: Category and Representations</h4>

<a id="X86EE897086995E47" name="X86EE897086995E47"></a>

<h4>10.2 Functors: Constructors</h4>

<a id="X7A21845C7C536717" name="X7A21845C7C536717"></a>

<h4>10.3 Functors: Attributes</h4>

<a id="X7D83D0EB87D2D872" name="X7D83D0EB87D2D872"></a>

<h4>10.4 Basic Functors</h4>

<a id="X7B9FE8BF80D47B6E" name="X7B9FE8BF80D47B6E"></a>

<h5>10.4-1 functor_Cokernel</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ functor_Cokernel</code></td><td class="tdright">( global variable )</td></tr></table></div>
The functor that associates to a map its cokernel.

<div class="example"><pre>
BindGlobal( "functor_Cokernel_for_fp_modules",
 CreateHomalgFunctor(
 [ "name", "Cokernel" ],
 [ "category", HOMALG_MODULES.category ],
 [ "operation", "Cokernel" ],
 [ "natural_transformation", "CokernelEpi" ],
 [ "special", true ],
 [ "number_of_arguments", 1 ],
 [ "1", [ [ "covariant" ],
 [ IsMapOfFinitelyGeneratedModulesRep,
 [ IsHomalgChainMorphism, IsImageSquare ] ] ] ],
 [ "OnObjects", _Functor_Cokernel_OnModules ]
 )
 );
</pre></div>

<a id="X875F177A82BF9B8B" name="X875F177A82BF9B8B"></a>

<h5>10.4-2 Cokernel</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cokernel</code>( <var class="Arg">phi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
The following example also makes use of the natural transformation <code class="code">CokernelEpi</code>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );;
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 3 generators>
gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz );;
gap> N := LeftPresentation( N );
<A non-torsion left module presented by 2 relations for 4 generators>
gap> mat := HomalgMatrix( "[ \
> 1, 0, -3, -6, \
> 0, 1, 6, 11, \
> 1, 0, -3, -6 \
> ]", 3, 4, zz );;
gap> phi := HomalgMap( mat, M, N );;
gap> IsMorphism( phi );
true
gap> phi;
<A homomorphism of left modules>
gap> coker := Cokernel( phi );
<A left module presented by 5 relations for 4 generators>
gap> ByASmallerPresentation( coker );
<A rank 1 left module presented by 1 relation for 2 generators>
gap> Display( coker );
Z/< 8 > + Z^(1 x 1)
gap> nu := CokernelEpi( phi );
<An epimorphism of left modules>
gap> Display( nu );
[ [ -5, 0 ],
 [ -6, 1 ],
 [ 1, -2 ],
 [ 0, 1 ] ]

the map is currently represented by the above 4 x 2 matrix
gap> DefectOfExactness( phi, nu );
<A zero left module>
gap> ByASmallerPresentation( nu );
<A non-zero epimorphism of left modules>
gap> Display( nu );
[ [ 2, 0 ],
 [ 1, -2 ],
 [ 0, 1 ] ]

the map is currently represented by the above 3 x 2 matrix
gap> PreInverse( nu );
false
</pre></div>

<a id="X7A5B3B307B334706" name="X7A5B3B307B334706"></a>

<h5>10.4-3 functor_ImageObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ functor_ImageObject</code></td><td class="tdright">( global variable )</td></tr></table></div>
The functor that associates to a map its image.

<div class="example"><pre>
BindGlobal( "functor_ImageObject_for_fp_modules",
 CreateHomalgFunctor(
 [ "name", "ImageObject for modules" ],
 [ "category", HOMALG_MODULES.category ],
 [ "operation", "ImageObject" ],
 [ "natural_transformation", "ImageObjectEmb" ],
 [ "number_of_arguments", 1 ],
 [ "1", [ [ "covariant" ],
 [ IsMapOfFinitelyGeneratedModulesRep and
 AdmissibleInputForHomalgFunctors ] ] ],
 [ "OnObjects", _Functor_ImageObject_OnModules ]
 )
 );
</pre></div>

<a id="X7E3FF900821DCBE6" name="X7E3FF900821DCBE6"></a>

<h5>10.4-4 ImageObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImageObject</code>( <var class="Arg">phi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
The following example also makes use of the natural transformations <code class="code">ImageObjectEpi</code> and <code class="code">ImageObjectEmb</code>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );;
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 3 generators>
gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz );;
gap> N := LeftPresentation( N );
<A non-torsion left module presented by 2 relations for 4 generators>
gap> mat := HomalgMatrix( "[ \
> 1, 0, -3, -6, \
> 0, 1, 6, 11, \
> 1, 0, -3, -6 \
> ]", 3, 4, zz );;
gap> phi := HomalgMap( mat, M, N );;
gap> IsMorphism( phi );
true
gap> phi;
<A homomorphism of left modules>
gap> im := ImageObject( phi );
<A left module presented by yet unknown relations for 3 generators>
gap> ByASmallerPresentation( im );
<A free left module of rank 1 on a free generator>
gap> pi := ImageObjectEpi( phi );
<A non-zero split epimorphism of left modules>
gap> epsilon := ImageObjectEmb( phi );
<A monomorphism of left modules>
gap> phi = pi * epsilon;
true
</pre></div>

<a id="X85C128B37E76827F" name="X85C128B37E76827F"></a>

<h5>10.4-5 Kernel</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Kernel</code>( <var class="Arg">phi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
The following example also makes use of the natural transformation <code class="code">KernelEmb</code>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );;
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 3 generators>
gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz );;
gap> N := LeftPresentation( N );
<A non-torsion left module presented by 2 relations for 4 generators>
gap> mat := HomalgMatrix( "[ \
> 1, 0, -3, -6, \
> 0, 1, 6, 11, \
> 1, 0, -3, -6 \
> ]", 3, 4, zz );;
gap> phi := HomalgMap( mat, M, N );;
gap> IsMorphism( phi );
true
gap> phi;
<A homomorphism of left modules>
gap> ker := Kernel( phi );
<A cyclic left module presented by yet unknown relations for a cyclic generato\
r>
gap> Display( ker );
Z/< -3 >
gap> ByASmallerPresentation( last );
<A cyclic torsion left module presented by 1 relation for a cyclic generator>
gap> Display( ker );
Z/< 3 >
gap> iota := KernelEmb( phi );
<A monomorphism of left modules>
gap> Display( iota );
[ [ 0, 1, 2 ] ]

the map is currently represented by the above 1 x 3 matrix
gap> DefectOfExactness( iota, phi );
<A zero left module>
gap> ByASmallerPresentation( iota );
<A non-zero monomorphism of left modules>
gap> Display( iota );
[ [ 1, 0 ] ]

the map is currently represented by the above 1 x 2 matrix
gap> PostInverse( iota );
<A non-zero split epimorphism of left modules>
</pre></div>

<a id="X7E6CDE7E85F09122" name="X7E6CDE7E85F09122"></a>

<h5>10.4-6 DefectOfExactness</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DefectOfExactness</code>( <var class="Arg">phi</var>, <var class="Arg">psi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
We follow the associative convention for applying maps. For left modules <var class="Arg">phi</var> is applied first and from the right. For right modules <var class="Arg">psi</var> is applied first and from the left.

The following example also makes use of the natural transformation <code class="code">KernelEmb</code>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 0, 5, 6, 7, 0 ]", 2, 4, zz );;
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 4 generators>
gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz );;
gap> N := LeftPresentation( N );
<A non-torsion left module presented by 2 relations for 4 generators>
gap> mat := HomalgMatrix( "[ \
> 1, 3, 3, 3, \
> 0, 3, 10, 17, \
> 1, 3, 3, 3, \
> 0, 0, 0, 0 \
> ]", 4, 4, zz );;
gap> phi := HomalgMap( mat, M, N );;
gap> IsMorphism( phi );
true
gap> phi;
<A homomorphism of left modules>
gap> iota := KernelEmb( phi );
<A monomorphism of left modules>
gap> DefectOfExactness( iota, phi );
<A zero left module>
gap> hom_iota := Hom( iota ); ## a shorthand for Hom( iota, zz );
<A homomorphism of right modules>
gap> hom_phi := Hom( phi ); ## a shorthand for Hom( phi, zz );
<A homomorphism of right modules>
gap> DefectOfExactness( hom_iota, hom_phi );
<A cyclic right module on a cyclic generator satisfying yet unknown relations>
gap> ByASmallerPresentation( last );
<A cyclic torsion right module on a cyclic generator satisfying 1 relation>
gap> Display( last );
Z/< 2 >
</pre></div>

<a id="X7B93718087EFD69B" name="X7B93718087EFD69B"></a>

<h5>10.4-7 Functor_Hom</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_Hom</code></td><td class="tdright">( global variable )</td></tr></table></div>
The bifunctor <code class="code">Hom</code>.

<div class="example"><pre>
BindGlobal( "Functor_Hom_for_fp_modules",
 CreateHomalgFunctor(
 [ "name", "Hom" ],
 [ "category", HOMALG_MODULES.category ],
 [ "operation", "Hom" ],
 [ "number_of_arguments", 2 ],
 [ "1", [ [ "contravariant", "right adjoint", "distinguished" ] ] ],
 [ "2", [ [ "covariant", "left exact" ] ] ],
 [ "OnObjects", _Functor_Hom_OnModules ],
 [ "OnMorphisms", _Functor_Hom_OnMaps ],
 [ "MorphismConstructor", HOMALG_MODULES.category.MorphismConstructor ]
 )
 );
</pre></div>

<a id="X80015C78876B4F1E" name="X80015C78876B4F1E"></a>

<h5>10.4-8 Hom</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Hom</code>( <var class="Arg">o1</var>, <var class="Arg">o2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<var class="Arg">o1</var> resp. <var class="Arg">o2</var> could be a module, a map, a complex (of modules or of again of complexes), or a chain morphism.

Each generator of a module of homomorphisms is displayed as a matrix of appropriate dimensions.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );;
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 3 generators>
gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz );;
gap> N := LeftPresentation( N );
<A non-torsion left module presented by 2 relations for 4 generators>
gap> mat := HomalgMatrix( "[ \
> 1, 0, -3, -6, \
> 0, 1, 6, 11, \
> 1, 0, -3, -6 \
> ]", 3, 4, zz );;
gap> phi := HomalgMap( mat, M, N );;
gap> IsMorphism( phi );
true
gap> phi;
<A homomorphism of left modules>
gap> psi := Hom( phi, M );
<A homomorphism of right modules>
gap> ByASmallerPresentation( psi );
<A non-zero homomorphism of right modules>
gap> Display( psi );
[ [ 1, 1, 1, 0 ],
 [ 2, 2, 0, 0 ],
 [ 0, 0, -2, 0 ] ]

the map is currently represented by the above 3 x 4 matrix
gap> homNM := Source( psi );
<A rank 2 right module on 4 generators satisfying 2 relations>
gap> IsIdenticalObj( homNM, Hom( N, M ) ); ## the caching at work
true
gap> homMM := Range( psi );
<A rank 1 right module on 3 generators satisfying 2 relations>
gap> IsIdenticalObj( homMM, Hom( M, M ) ); ## the caching at work
true
gap> Display( homNM );
Z/< 3 > + Z/< 3 > + Z^(2 x 1)
gap> Display( homMM );
Z/< 3 > + Z/< 3 > + Z^(1 x 1)
gap> IsMonomorphism( psi );
false
gap> IsEpimorphism( psi );
false
gap> GeneratorsOfModule( homMM );
<A set of 3 generators of a homalg right module>
gap> Display( last );
[ [ 0, 0, 0 ],
 [ 0, 1, 2 ],
 [ 0, 0, 0 ] ]

the map is currently represented by the above 3 x 3 matrix

[ [ 0, 2, 4 ],
 [ 0, 0, 0 ],
 [ 0, 2, 4 ] ]

the map is currently represented by the above 3 x 3 matrix

[ [ 0, 0, 1 ],
 [ 0, 2, 2 ],
 [ 0, 0, 1 ] ]

the map is currently represented by the above 3 x 3 matrix

a set of 3 generators given by the the above matrices
gap> GeneratorsOfModule( homNM );
<A set of 4 generators of a homalg right module>
gap> Display( last );
[ [ 0, 1, 2 ],
 [ 0, 1, 2 ],
 [ 0, 1, 2 ],
 [ 0, 0, 0 ] ]

the map is currently represented by the above 4 x 3 matrix

[ [ 0, 1, 2 ],
 [ 0, 0, 0 ],
 [ 0, 0, 0 ],
 [ 0, 2, 4 ] ]

the map is currently represented by the above 4 x 3 matrix

[ [ 0, 0, 1 ],
 [ 0, 1, -1 ],
 [ 0, 1, 5 ],
 [ 0, 1, 1 ] ]

the map is currently represented by the above 4 x 3 matrix

[ [ 0, 0, 0 ],
 [ 0, 0, 1 ],
 [ 0, 0, -2 ],
 [ 0, 0, 1 ] ]

the map is currently represented by the above 4 x 3 matrix

a set of 4 generators given by the the above matrices
</pre></div>

If for example the source N gets a new presentation, you will see the effect on the generators:

<div class="example"><pre>
gap> ByASmallerPresentation( N );
<A rank 2 left module presented by 1 relation for 3 generators>
gap> GeneratorsOfModule( homNM );
<A set of 4 generators of a homalg right module>
gap> Display( last );
[ [ 0, 3, 6 ],
 [ 0, 1, 2 ],
 [ 0, 0, 0 ] ]

the map is currently represented by the above 3 x 3 matrix

[ [ 0, 9, 18 ],
 [ 0, 0, 0 ],
 [ 0, 2, 4 ] ]

the map is currently represented by the above 3 x 3 matrix

[ [ 0, 9, 18 ],
 [ 0, 1, 5 ],
 [ 0, 1, 1 ] ]

the map is currently represented by the above 3 x 3 matrix

[ [ 0, 0, 0 ],
 [ 0, 0, -2 ],
 [ 0, 0, 1 ] ]

the map is currently represented by the above 3 x 3 matrix

a set of 4 generators given by the the above matrices
</pre></div>

Now we compute a certain natural filtration on <code class="code">Hom</code>(M,M):

<div class="example"><pre>
gap> dM := Resolution( M );
<A non-zero right acyclic complex containing a single morphism of left modules\
at degrees [ 0 .. 1 ]>
gap> hMM := Hom( dM, dM );
<A non-zero acyclic cocomplex containing a single morphism of right complexes \
at degrees [ 0 .. 1 ]>
gap> BMM := HomalgBicomplex( hMM );
<A non-zero bicocomplex containing right modules at bidegrees [ 0 .. 1 ]x
[ -1 .. 0 ]>
gap> II_E := SecondSpectralSequenceWithFiltration( BMM );
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of right 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 );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:

-1: <A non-zero cyclic torsion right module on a cyclic generator satisfying
 yet unknown relations>
 0: <A rank 1 right module on 3 generators satisfying 2 relations>
of
<A right module on 4 generators satisfying yet unknown relations>>
gap> ByASmallerPresentation( filt );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:

-1: <A non-zero cyclic torsion right module on a cyclic generator satisfying 1\
relation>
 0: <A rank 1 right module on 2 generators satisfying 1 relation>
of
<A rank 1 right module on 3 generators satisfying 2 relations>>
gap> Display( filt );
Degree -1:

Z/< 3 >
----------
Degree 0:

Z/< 3 > + Z^(1 x 1)
gap> Display( homMM );
Z/< 3 > + Z/< 3 > + Z^(1 x 1)
</pre></div>

<a id="X7A1A077D8268FADE" name="X7A1A077D8268FADE"></a>

<h5>10.4-9 Functor_TensorProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_TensorProduct</code></td><td class="tdright">( global variable )</td></tr></table></div>
The tensor product bifunctor.

<div class="example"><pre>
BindGlobal( "Functor_TensorProduct_for_fp_modules",
 CreateHomalgFunctor(
 [ "name", "TensorProduct" ],
 [ "category", HOMALG_MODULES.category ],
 [ "operation", "TensorProductOp" ],
 [ "number_of_arguments", 2 ],
 [ "1", [ [ "covariant", "left adjoint", "distinguished" ] ] ],
 [ "2", [ [ "covariant", "left adjoint" ] ] ],
 [ "OnObjects", _Functor_TensorProduct_OnModules ],
 [ "OnMorphisms", _Functor_TensorProduct_OnMaps ],
 [ "MorphismConstructor", HOMALG_MODULES.category.MorphismConstructor ]
 )
 );
</pre></div>

<a id="X87EB0B4A852CF4C6" name="X87EB0B4A852CF4C6"></a>

<h5>10.4-10 TensorProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProduct</code>( <var class="Arg">o1</var>, <var class="Arg">o2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \*</code>( <var class="Arg">o1</var>, <var class="Arg">o2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<var class="Arg">o1</var> resp. <var class="Arg">o2</var> could be a module, a map, a complex (of modules or of again of complexes), or a chain morphism.

The symbol <code class="code">*</code> is a shorthand for several operations associated with the functor <code class="code">Functor_TensorProduct_for_fp_modules</code> installed under the name <code class="code">TensorProduct</code>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );
<A 2 x 3 matrix over an internal ring>
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 3 generators>
gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz );
<A 2 x 4 matrix over an internal ring>
gap> N := LeftPresentation( N );
<A non-torsion left module presented by 2 relations for 4 generators>
gap> mat := HomalgMatrix( "[ \
> 1, 0, -3, -6, \
> 0, 1, 6, 11, \
> 1, 0, -3, -6 \
> ]", 3, 4, zz );
<A 3 x 4 matrix over an internal ring>
gap> phi := HomalgMap( mat, M, N );
<A "homomorphism" of left modules>
gap> IsMorphism( phi );
true
gap> phi;
<A homomorphism of left modules>
gap> L := Hom( zz, M );
<A rank 1 right module on 3 generators satisfying yet unknown relations>
gap> ByASmallerPresentation( L );
<A rank 1 right module on 2 generators satisfying 1 relation>
gap> Display( L );
Z/< 3 > + Z^(1 x 1)
gap> L;
<A rank 1 right module on 2 generators satisfying 1 relation>
gap> psi := phi * L;
<A homomorphism of right modules>
gap> ByASmallerPresentation( psi );
<A non-zero homomorphism of right modules>
gap> Display( psi );
[ [ 0, 0, 1, 1 ],
 [ 0, 0, 8, 1 ],
 [ 0, 0, 0, -2 ],
 [ 0, 0, 0, 2 ] ]

the map is currently represented by the above 4 x 4 matrix
gap> ML := Source( psi );
<A rank 1 right module on 4 generators satisfying 3 relations>
gap> IsIdenticalObj( ML, M * L ); ## the caching at work
true
gap> NL := Range( psi );
<A rank 2 right module on 4 generators satisfying 2 relations>
gap> IsIdenticalObj( NL, N * L ); ## the caching at work
true
gap> Display( ML );
Z/< 3 > + Z/< 3 > + Z/< 3 > + Z^(1 x 1)
gap> Display( NL );
Z/< 3 > + Z/< 12 > + Z^(2 x 1)
</pre></div>

Now we compute a certain natural filtration on the tensor product M<code class="code">*</code>L:

<div class="example"><pre>
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> CE := Resolution( GP );
<An acyclic cocomplex containing a single morphism of right complexes at degre\
es [ 0 .. 1 ]>
gap> FCE := Hom( CE, L );
<A non-zero acyclic complex containing a single morphism of left cocomplexes a\
t degrees [ 0 .. 1 ]>
gap> BC := HomalgBicomplex( FCE );
<A non-zero bicomplex containing left modules at bidegrees [ 0 .. 1 ]x
[ -1 .. 0 ]>
gap> II_E := SecondSpectralSequenceWithFiltration( BC );
<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 );
<An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A rank 1 left module presented by 1 relation for 2 generators>
 -1: <A non-zero left module presented by 2 relations for 2 generators>
of
<A non-zero left module presented by 4 relations for 4 generators>>
gap> ByASmallerPresentation( filt );
<An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A rank 1 left module presented by 1 relation for 2 generators>
 -1: <A non-zero torsion left module presented by 2 relations
 for 2 generators>
of
<A rank 1 left module presented by 3 relations for 4 generators>>
gap> Display( filt );
Degree 0:

Z/< 3 > + Z^(1 x 1)
----------
Degree -1:

Z/< 3 > + Z/< 3 >
gap> Display( ML );
Z/< 3 > + Z/< 3 > + Z/< 3 > + Z^(1 x 1)
</pre></div>

<a id="X7D007A7079F7BEE3" name="X7D007A7079F7BEE3"></a>

<h5>10.4-11 Functor_Ext</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_Ext</code></td><td class="tdright">( global variable )</td></tr></table></div>
The bifunctor <code class="code">Ext</code>.

Below is the only specific line of code used to define <code class="code">Functor_Ext_for_fp_modules</code> and all the different operations <code class="code">Ext</code> in homalg.

<div class="example"><pre>
RightSatelliteOfCofunctor( Functor_Hom_for_fp_modules, "Ext" );
</pre></div>

<a id="X8692578881E71913" name="X8692578881E71913"></a>

<h5>10.4-12 Ext</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ext</code>( [<var class="Arg">c</var>, ]<var class="Arg">o1</var>, <var class="Arg">o2</var>[, <var class="Arg">str</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Compute the <var class="Arg">c</var>-th extension object of <var class="Arg">o1</var> with <var class="Arg">o2</var> where <var class="Arg">c</var> is a nonnegative integer and <var class="Arg">o1</var> resp. <var class="Arg">o2</var> could be a module, a map, a complex (of modules or of again of complexes), or a chain morphism. If <var class="Arg">str</var>=<q>a</q> then the (cohomologically) graded object Ext^i(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤<var class="Arg">c</var> is computed. If neither <var class="Arg">c</var> nor <var class="Arg">str</var> is specified then the cohomologically graded object Ext^i(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤ d is computed, where d is the length of the internally computed free resolution of <var class="Arg">o1</var>.

Each generator of a module of extensions is displayed as a matrix of appropriate dimensions.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );;
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 3 generators>
gap> N := TorsionObject( M );
<A cyclic torsion left module presented by yet unknown relations for a cyclic \
generator>
gap> iota := TorsionObjectEmb( M );
<A monomorphism of left modules>
gap> psi := Ext( 1, iota, N );
<A homomorphism of right modules>
gap> ByASmallerPresentation( psi );
<A non-zero homomorphism of right modules>
gap> Display( psi );
[ [ 1 ] ]

the map is currently represented by the above 1 x 1 matrix
gap> extNN := Range( psi );
<A non-zero cyclic torsion right module on a cyclic generator satisfying 1 rel\
ation>
gap> IsIdenticalObj( extNN, Ext( 1, N, N ) ); ## the caching at work
true
gap> extMN := Source( psi );
<A non-zero cyclic torsion right module on a cyclic generator satisfying 1 rel\
ation>
gap> IsIdenticalObj( extMN, Ext( 1, M, N ) ); ## the caching at work
true
gap> Display( extNN );
Z/< 3 >
gap> Display( extMN );
Z/< 3 >
</pre></div>

<a id="X821034FE80907E8D" name="X821034FE80907E8D"></a>

<h5>10.4-13 Functor_Tor</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_Tor</code></td><td class="tdright">( global variable )</td></tr></table></div>
The bifunctor <code class="code">Tor</code>.

Below is the only specific line of code used to define <code class="code">Functor_Tor_for_fp_modules</code> and all the different operations <code class="code">Tor</code> in homalg.

<div class="example"><pre>
LeftSatelliteOfFunctor( Functor_TensorProduct_for_fp_modules, "Tor" );
</pre></div>

<a id="X79821906875CF49E" name="X79821906875CF49E"></a>

<h5>10.4-14 Tor</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tor</code>( [<var class="Arg">c</var>, ]<var class="Arg">o1</var>, <var class="Arg">o2</var>[, <var class="Arg">str</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Compute the <var class="Arg">c</var>-th torsion object of <var class="Arg">o1</var> with <var class="Arg">o2</var> where <var class="Arg">c</var> is a nonnegative integer and <var class="Arg">o1</var> resp. <var class="Arg">o2</var> could be a module, a map, a complex (of modules or of again of complexes), or a chain morphism. If <var class="Arg">str</var>=<q>a</q> then the (cohomologically) graded object Tor_i(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤<var class="Arg">c</var> is computed. If neither <var class="Arg">c</var> nor <var class="Arg">str</var> is specified then the cohomologically graded object Tor_i(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤ d is computed, where d is the length of the internally computed free resolution of <var class="Arg">o1</var>.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz );;
gap> M := LeftPresentation( M );
<A non-torsion left module presented by 2 relations for 3 generators>
gap> N := TorsionObject( M );
<A cyclic torsion left module presented by yet unknown relations for a cyclic \
generator>
gap> iota := TorsionObjectEmb( M );
<A monomorphism of left modules>
gap> psi := Tor( 1, iota, N );
<A homomorphism of left modules>
gap> ByASmallerPresentation( psi );
<A non-zero homomorphism of left modules>
gap> Display( psi );
[ [ 1 ] ]

the map is currently represented by the above 1 x 1 matrix
gap> torNN := Source( psi );
<A non-zero cyclic torsion left module presented by 1 relation for a cyclic ge\
nerator>
gap> IsIdenticalObj( torNN, Tor( 1, N, N ) ); ## the caching at work
true
gap> torMN := Range( psi );
<A non-zero cyclic torsion left module presented by 1 relation for a cyclic ge\
nerator>
gap> IsIdenticalObj( torMN, Tor( 1, M, N ) ); ## the caching at work
true
gap> Display( torNN );
Z/< 3 >
gap> Display( torMN );
Z/< 3 >
</pre></div>

<a id="X84C60D997A79524E" name="X84C60D997A79524E"></a>

<h5>10.4-15 Functor_RHom</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_RHom</code></td><td class="tdright">( global variable )</td></tr></table></div>
The bifunctor <code class="code">RHom</code>.

Below is the only specific line of code used to define <code class="code">Functor_RHom_for_fp_modules</code> and all the different operations <code class="code">RHom</code> in homalg.

<div class="example"><pre>
RightDerivedCofunctor( Functor_Hom_for_fp_modules );
</pre></div>

<a id="X7D8BDC0C817C10AB" name="X7D8BDC0C817C10AB"></a>

<h5>10.4-16 RHom</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RHom</code>( [<var class="Arg">c</var>, ]<var class="Arg">o1</var>, <var class="Arg">o2</var>[, <var class="Arg">str</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Compute the <var class="Arg">c</var>-th extension object of <var class="Arg">o1</var> with <var class="Arg">o2</var> where <var class="Arg">c</var> is a nonnegative integer and <var class="Arg">o1</var> resp. <var class="Arg">o2</var> could be a module, a map, a complex (of modules or of again of complexes), or a chain morphism. The string <var class="Arg">str</var> may take different values:

<ul>
<li>If <var class="Arg">str</var>=<q>a</q> then R^i Hom(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤<var class="Arg">c</var> is computed.

</li>
<li>If <var class="Arg">str</var>=<q>c</q> then the <var class="Arg">c</var>-th connecting homomorphism with respect to the short exact sequence <var class="Arg">o1</var> is computed.

</li>
<li>If <var class="Arg">str</var>=<q>t</q> then the exact triangle upto cohomological degree <var class="Arg">c</var> with respect to the short exact sequence <var class="Arg">o1</var> is computed.

</li>
</ul>
If neither <var class="Arg">c</var> nor <var class="Arg">str</var> is specified then the cohomologically graded object R^i Hom(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤ d is computed, where d is the length of the internally computed free resolution of <var class="Arg">o1</var>.

Each generator of a module of derived homomorphisms is displayed as a matrix of appropriate dimensions.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> m := HomalgMatrix( [ [ 8, 0 ], [ 0, 2 ] ], zz );;
gap> M := LeftPresentation( m );
<A left module presented by 2 relations for 2 generators>
gap> Display( M );
Z/< 8 > + Z/< 2 >
gap> M;
<A torsion left module presented by 2 relations for 2 generators>
gap> a := HomalgMatrix( [ [ 2, 0 ] ], zz );;
gap> alpha := HomalgMap( a, "free", M );
<A homomorphism of left modules>
gap> pi := CokernelEpi( alpha );
<An epimorphism of left modules>
gap> Display( pi );
[ [ 1, 0 ],
 [ 0, 1 ] ]

the map is currently represented by the above 2 x 2 matrix
gap> iota := KernelEmb( pi );
<A monomorphism of left modules>
gap> Display( iota );
[ [ 2, 0 ] ]

the map is currently represented by the above 1 x 2 matrix
gap> N := Kernel( pi );
<A cyclic torsion left module presented by yet unknown relations for a cyclic \
generator>
gap> Display( N );
Z/< 4 >
gap> C := HomalgComplex( pi );
<A left acyclic complex containing a single morphism of left modules at degree\
s [ 0 .. 1 ]>
gap> Add( C, iota );
gap> ByASmallerPresentation( C );
<A non-zero short exact sequence containing
2 morphisms of left modules at degrees [ 0 .. 2 ]>
gap> Display( C );
-------------------------
at homology degree: 2
Z/< 4 >
-------------------------
[ [ 0, 6 ] ]

the map is currently represented by the above 1 x 2 matrix
------------v------------
at homology degree: 1
Z/< 2 > + Z/< 8 >
-------------------------
[ [ 0, 1 ],
 [ 1, 1 ] ]

the map is currently represented by the above 2 x 2 matrix
------------v------------
at homology degree: 0
Z/< 2 > + Z/< 2 >
-------------------------
gap> T := RHom( C, N );
<An exact cotriangle containing 3 morphisms of right cocomplexes at degrees
[ 0, 1, 2, 0 ]>
gap> ByASmallerPresentation( T );
<A non-zero exact cotriangle containing
3 morphisms of right cocomplexes at degrees [ 0, 1, 2, 0 ]>
gap> L := LongSequence( T );
<A cosequence containing 5 morphisms of right modules at degrees [ 0 .. 5 ]>
gap> Display( L );
-------------------------
at cohomology degree: 5
Z/< 4 >
------------^------------
[ [ 0, 3 ] ]

the map is currently represented by the above 1 x 2 matrix
-------------------------
at cohomology degree: 4
Z/< 2 > + Z/< 4 >
------------^------------
[ [ 0, 1 ],
 [ 0, 0 ] ]

the map is currently represented by the above 2 x 2 matrix
-------------------------
at cohomology degree: 3
Z/< 2 > + Z/< 2 >
------------^------------
[ [ 1 ],
 [ 0 ] ]

the map is currently represented by the above 2 x 1 matrix
-------------------------
at cohomology degree: 2
Z/< 4 >
------------^------------
[ [ 0, 2 ] ]

the map is currently represented by the above 1 x 2 matrix
-------------------------
at cohomology degree: 1
Z/< 2 > + Z/< 4 >
------------^------------
[ [ 0, 1 ],
 [ 2, 2 ] ]

the map is currently represented by the above 2 x 2 matrix
-------------------------
at cohomology degree: 0
Z/< 2 > + Z/< 2 >
-------------------------
gap> IsExactSequence( L );
true
gap> L;
<An exact cosequence containing 5 morphisms of right modules at degrees
[ 0 .. 5 ]>
</pre></div>

<a id="X806251E3836C00B9" name="X806251E3836C00B9"></a>

<h5>10.4-17 Functor_LTensorProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_LTensorProduct</code></td><td class="tdright">( global variable )</td></tr></table></div>
The bifunctor <code class="code">LTensorProduct</code>.

Below is the only specific line of code used to define <code class="code">Functor_LTensorProduct_for_fp_modules</code> and all the different operations <code class="code">LTensorProduct</code> in homalg.

<div class="example"><pre>
LeftDerivedFunctor( Functor_TensorProduct_for_fp_modules );
</pre></div>

<a id="X7C12DA648798E77E" name="X7C12DA648798E77E"></a>

<h5>10.4-18 LTensorProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LTensorProduct</code>( [<var class="Arg">c</var>, ]<var class="Arg">o1</var>, <var class="Arg">o2</var>[, <var class="Arg">str</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Compute the <var class="Arg">c</var>-th torsion object of <var class="Arg">o1</var> with <var class="Arg">o2</var> where <var class="Arg">c</var> is a nonnegative integer and <var class="Arg">o1</var> resp. <var class="Arg">o2</var> could be a module, a map, a complex (of modules or of again of complexes), or a chain morphism. The string <var class="Arg">str</var> may take different values:

<ul>
<li>If <var class="Arg">str</var>=<q>a</q> then L_i TensorProduct(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤<var class="Arg">c</var> is computed.

</li>
<li>If <var class="Arg">str</var>=<q>c</q> then the <var class="Arg">c</var>-th connecting homomorphism with respect to the short exact sequence <var class="Arg">o1</var> is computed.

</li>
<li>If <var class="Arg">str</var>=<q>t</q> then the exact triangle upto cohomological degree <var class="Arg">c</var> with respect to the short exact sequence <var class="Arg">o1</var> is computed.

</li>
</ul>
If neither <var class="Arg">c</var> nor <var class="Arg">str</var> is specified then the cohomologically graded object L_i TensorProduct(<var class="Arg">o1</var>,<var class="Arg">o2</var>) for 0 ≤ i ≤ d is computed, where d is the length of the internally computed free resolution of <var class="Arg">o1</var>.

Each generator of a module of derived homomorphisms is displayed as a matrix of appropriate dimensions.

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> m := HomalgMatrix( [ [ 8, 0 ], [ 0, 2 ] ], zz );;
gap> M := LeftPresentation( m );
<A left module presented by 2 relations for 2 generators>
gap> Display( M );
Z/< 8 > + Z/< 2 >
gap> M;
<A torsion left module presented by 2 relations for 2 generators>
gap> a := HomalgMatrix( [ [ 2, 0 ] ], zz );;
gap> alpha := HomalgMap( a, "free", M );
<A homomorphism of left modules>
gap> pi := CokernelEpi( alpha );
<An epimorphism of left modules>
gap> Display( pi );
[ [ 1, 0 ],
 [ 0, 1 ] ]

the map is currently represented by the above 2 x 2 matrix
gap> iota := KernelEmb( pi );
<A monomorphism of left modules>
gap> Display( iota );
[ [ 2, 0 ] ]

the map is currently represented by the above 1 x 2 matrix
gap> N := Kernel( pi );
<A cyclic torsion left module presented by yet unknown relations for a cyclic \
generator>
gap> Display( N );
Z/< 4 >
gap> C := HomalgComplex( pi );
<A left acyclic complex containing a single morphism of left modules at degree\
s [ 0 .. 1 ]>
gap> Add( C, iota );
gap> ByASmallerPresentation( C );
<A non-zero short exact sequence containing
2 morphisms of left modules at degrees [ 0 .. 2 ]>
gap> Display( C );
-------------------------
at homology degree: 2
Z/< 4 >
-------------------------
[ [ 0, 6 ] ]

the map is currently represented by the above 1 x 2 matrix
------------v------------
at homology degree: 1
Z/< 2 > + Z/< 8 >
-------------------------
[ [ 0, 1 ],
 [ 1, 1 ] ]

the map is currently represented by the above 2 x 2 matrix
------------v------------
at homology degree: 0
Z/< 2 > + Z/< 2 >
-------------------------
gap> T := LTensorProduct( C, N );
<An exact triangle containing 3 morphisms of left complexes at degrees
[ 1, 2, 3, 1 ]>
gap> ByASmallerPresentation( T );
<A non-zero exact triangle containing
3 morphisms of left complexes at degrees [ 1, 2, 3, 1 ]>
gap> L := LongSequence( T );
<A sequence containing 5 morphisms of left modules at degrees [ 0 .. 5 ]>
gap> Display( L );
-------------------------
at homology degree: 5
Z/< 4 >
-------------------------
[ [ 0, 3 ] ]

the map is currently represented by the above 1 x 2 matrix
------------v------------
at homology degree: 4
Z/< 2 > + Z/< 4 >
-------------------------
[ [ 0, 1 ],
 [ 0, 0 ] ]

the map is currently represented by the above 2 x 2 matrix
------------v------------
at homology degree: 3
Z/< 2 > + Z/< 2 >
-------------------------
[ [ 2 ],
 [ 0 ] ]

the map is currently represented by the above 2 x 1 matrix
------------v------------
at homology degree: 2
Z/< 4 >
-------------------------
[ [ 0, 2 ] ]

the map is currently represented by the above 1 x 2 matrix
------------v------------
at homology degree: 1
Z/< 2 > + Z/< 4 >
-------------------------
[ [ 0, 1 ],
 [ 1, 1 ] ]

the map is currently represented by the above 2 x 2 matrix
------------v------------
at homology degree: 0
Z/< 2 > + Z/< 2 >
-------------------------
gap> IsExactSequence( L );
true
gap> L;
<An exact sequence containing 5 morphisms of left modules at degrees
[ 0 .. 5 ]>
</pre></div>

<a id="X7ACC6A7C86E4354C" name="X7ACC6A7C86E4354C"></a>

<h5>10.4-19 Functor_HomHom</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_HomHom</code></td><td class="tdright">( global variable )</td></tr></table></div>
The bifunctor <code class="code">HomHom</code>.

Below is the only specific line of code used to define <code class="code">Functor_HomHom_for_fp_modules</code> and all the different operations <code class="code">HomHom</code> in homalg.

<div class="example"><pre>
Functor_Hom_for_fp_modules * Functor_Hom_for_fp_modules;
</pre></div>

<a id="X84557A6B79382720" name="X84557A6B79382720"></a>

<h5>10.4-20 Functor_LHomHom</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Functor_LHomHom</code></td><td class="tdright">( global variable )</td></tr></table></div>
The bifunctor <code class="code">LHomHom</code>.

Below is the only specific line of code used to define <code class="code">Functor_LHomHom_for_fp_modules</code> and all the different operations <code class="code">LHomHom</code> in homalg.

<div class="example"><pre>
LeftDerivedFunctor( Functor_HomHom_for_fp_modules );
</pre></div>

<a id="X815BF6DA7FD5D44B" name="X815BF6DA7FD5D44B"></a>

<h4>10.5 Tool Functors</h4>

<a id="X879135AC8330C509" name="X879135AC8330C509"></a>

<h4>10.6 Other Functors</h4>

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<h4>10.7 Functors: Operations and Functions</h4>

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