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1 Categories of finitely presented graded modules
 1.1 Constructors

  1.1-1 LeftPresentationWithDegrees
  1.1-2 RightPresentationWithDegrees
  1.1-3 CategoryOfHomalgFinitelyPresentedGradedLeftModules
  1.1-4 CategoryOfHomalgFinitelyPresentedGradedRightModules
 1.2 Functors

  1.2-1 FunctorStandardGradedModuleLeft
  1.2-2 FunctorStandardGradedModuleRight
  1.2-3 FunctorGetRidOfZeroHomogeneousGeneratorsLeft
  1.2-4 FunctorGetRidOfZeroHomogeneousGeneratorsRight
  1.2-5 FunctorLessHomogeneousGeneratorsLeft
  1.2-6 FunctorLessHomogeneousGeneratorsRight
  1.2-7 FunctorGradedDualLeft
  1.2-8 FunctorGradedDualRight
  1.2-9 FunctorDoubleGradedDualLeft
  1.2-10 FunctorDoubleGradedDualRight
 1.3 Natural transformations

  1.3-1 NaturalIsomorphismFromIdentityToStandardGradedModuleLeft
  1.3-2 NaturalIsomorphismFromIdentityToStandardGradedModuleRight
  1.3-3 NaturalIsomorphismFromIdentityToGetRidOfZeroHomogeneousGeneratorsLeft
  1.3-4 NaturalIsomorphismFromIdentityToGetRidOfZeroHomogeneousGeneratorsRight
  1.3-5 NaturalIsomorphismFromIdentityToLessHomogeneousGeneratorsLeft
  1.3-6 NaturalIsomorphismFromIdentityToLessHomogeneousGeneratorsRight
  1.3-7 NaturalTransformationFromIdentityToDoubleGradedDualLeft
  1.3-8 NaturalTransformationFromIdentityToDoubleGradedDualRight
 1.4 Examples
 1.5 Supported CAP operations

  1.5-1 Category of of finitely presented graded modules
 1.6 GAP categories

  1.6-1 IsCategoryOfHomalgGradedModules

1 Categories of finitely presented graded modules

1.1 Constructors

1.1-1 LeftPresentationWithDegrees
‣ LeftPresentationWithDegrees( mat )( operation )

Returns: a homalg graded left module

This constructor returns the finitely presented left module with homogeneous relations given by the rows of the homalg matrix mat.

1.1-2 RightPresentationWithDegrees
‣ RightPresentationWithDegrees( mat )( operation )

Returns: a homalg graded right module

This constructor returns the finitely presented right module with homogeneous relations given by the columns of the homalg matrix mat.

1.1-3 CategoryOfHomalgFinitelyPresentedGradedLeftModules
‣ CategoryOfHomalgFinitelyPresentedGradedLeftModules( R )( attribute )

Returns: an intrinsic cateogry of left modules

Construct the category of finitely presented graded left modules over the computable graded ring S.

1.1-4 CategoryOfHomalgFinitelyPresentedGradedRightModules
‣ CategoryOfHomalgFinitelyPresentedGradedRightModules( R )( attribute )

Returns: an intrinsic cateogry of right modules

Construct the category of finitely presented graded right modules over the computable graded ring S.

1.2 Functors

1.2-1 FunctorStandardGradedModuleLeft
‣ FunctorStandardGradedModuleLeft( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\). The output is a functor which takes a graded left presentation as input and computes its standard presentation.

1.2-2 FunctorStandardGradedModuleRight
‣ FunctorStandardGradedModuleRight( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\). The output is a functor which takes a graded right presentation as input and computes its standard presentation.

1.2-3 FunctorGetRidOfZeroHomogeneousGeneratorsLeft
‣ FunctorGetRidOfZeroHomogeneousGeneratorsLeft( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\). The output is a functor which takes a graded left presentation as input and gets rid of the zero generators.

1.2-4 FunctorGetRidOfZeroHomogeneousGeneratorsRight
‣ FunctorGetRidOfZeroHomogeneousGeneratorsRight( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\). The output is a functor which takes a graded right presentation as input and gets rid of the zero generators.

1.2-5 FunctorLessHomogeneousGeneratorsLeft
‣ FunctorLessHomogeneousGeneratorsLeft( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\). The output is functor which takes a graded left presentation as input and computes a presentation having less generators.

1.2-6 FunctorLessHomogeneousGeneratorsRight
‣ FunctorLessHomogeneousGeneratorsRight( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\). The output is functor which takes a graded right presentation as input and computes a presentation having less generators.

1.2-7 FunctorGradedDualLeft
‣ FunctorGradedDualLeft( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\) that has an involution function. The output is functor which takes a graded left presentation M as input and computes its Hom(M, R) as a graded left presentation.

1.2-8 FunctorGradedDualRight
‣ FunctorGradedDualRight( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\) that has an involution function. The output is functor which takes a graded right presentation M as input and computes its Hom(M, R) as a graded right presentation.

1.2-9 FunctorDoubleGradedDualLeft
‣ FunctorDoubleGradedDualLeft( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\) that has an involution function. The output is functor which takes a graded left presentation M as input and computes its Hom( Hom(M, R), R ) as a graded left presentation.

1.2-10 FunctorDoubleGradedDualRight
‣ FunctorDoubleGradedDualRight( S )( attribute )

Returns: a functor

The argument is a homalg graded ring \(S\) that has an involution function. The output is functor which takes a graded right presentation M as input and computes its Hom( Hom(M, R), R ) as a graded right presentation.

1.3 Natural transformations

1.3-1 NaturalIsomorphismFromIdentityToStandardGradedModuleLeft
‣ NaturalIsomorphismFromIdentityToStandardGradedModuleLeft( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{StandardGradedModuleLeft}\)

The argument is a homalg graded ring \(S\). The output is the natural isomorphism from the identity functor to the left standard module functor.

1.3-2 NaturalIsomorphismFromIdentityToStandardGradedModuleRight
‣ NaturalIsomorphismFromIdentityToStandardGradedModuleRight( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{StandardGradedModuleRight}\)

The argument is a homalg graded ring \(S\). The output is the natural isomorphism from the identity functor to the right standard module functor.

1.3-3 NaturalIsomorphismFromIdentityToGetRidOfZeroHomogeneousGeneratorsLeft
‣ NaturalIsomorphismFromIdentityToGetRidOfZeroHomogeneousGeneratorsLeft( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroHomogeneousGeneratorsLeft}\)

The argument is a homalg graded ring \(S\). The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of left modules.

1.3-4 NaturalIsomorphismFromIdentityToGetRidOfZeroHomogeneousGeneratorsRight
‣ NaturalIsomorphismFromIdentityToGetRidOfZeroHomogeneousGeneratorsRight( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroHomogeneousGeneratorsRight}\)

The argument is a homalg graded ring \(S\). The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of right modules.

1.3-5 NaturalIsomorphismFromIdentityToLessHomogeneousGeneratorsLeft
‣ NaturalIsomorphismFromIdentityToLessHomogeneousGeneratorsLeft( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{LessHomogeneousGeneratorsLeft}\)

The argument is a homalg graded ring \(S\). The output is the natural morphism from the identity functor to the left less generators functor.

1.3-6 NaturalIsomorphismFromIdentityToLessHomogeneousGeneratorsRight
‣ NaturalIsomorphismFromIdentityToLessHomogeneousGeneratorsRight( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{LessHomogeneousGeneratorsRight}\)

The argument is a homalg graded ring \(S\). The output is the natural morphism from the identity functor to the right less generator functor.

1.3-7 NaturalTransformationFromIdentityToDoubleGradedDualLeft
‣ NaturalTransformationFromIdentityToDoubleGradedDualLeft( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{FunctorDoubleGradedDualLeft}\)

The argument is a homalg graded ring \(S\). The output is the natural morphism from the identity functor to the double dual functor in left Presentations category.

1.3-8 NaturalTransformationFromIdentityToDoubleGradedDualRight
‣ NaturalTransformationFromIdentityToDoubleGradedDualRight( S )( attribute )

Returns: a natural transformation \(\mathrm{Id} \rightarrow \mathrm{FunctorDoubleGradedDualRight}\)

The argument is a homalg graded ring \(S\). The output is the natural morphism from the identity functor to the double dual functor in right Presentations category.

1.4 Examples

gap> Q := GradedRing( HomalgFieldOfRationalsInSingular( ) );
Q
(weights: yet unset)
gap> S := Q["x,y,z"];
Q[x,y,z]
(weights: yet unset)
gap> A := CategoryOfHomalgFinitelyPresentedGradedLeftModules( S );
IntrinsicCategory( CategoryWithAmbientObjects( Freyd( GradedRows(
Q[x,y,z] (with weights [ 1, 1, 1 ]) ) ) ) )
gap> Display( A );
A CAP category with name
IntrinsicCategory( CategoryWithAmbientObjects(
Freyd( GradedRows( Q[x,y,z] (with weights [ 1, 1, 1 ]) ) ) ) ):

184 primitive operations were used to derive 376 operations for this category
which algorithmically
* IsEquippedWithHomomorphismStructure
* IsSymmetricClosedMonoidalCategory
* IsAbelianCategoryWithEnoughProjectives
gap> mat := HomalgMatrix( "[ 0, z, -y,  -z, 0, x,  y, -x, 0 ]", 3, 3, S );
<A 3 x 3 matrix over a graded ring>
gap> phi := GradedMap( mat, "left", S );
<A homomorphism of left graded modules>
gap> IsWellDefined( phi );
true
gap> phi;
<A homomorphism of left graded modules>
gap> N := CokernelObject( phi );
<A graded left module presented by 3 relations for 3 generators>
gap> Display( N );
0, z, -y,
-z,0, x,
y, -x,0
(over a graded ring)

Cokernel of the map

R^(1x3) --> R^(1x3), ( for R := Q[x,y,z] (with weights [ 1, 1, 1 ]) )

currently represented by the above matrix

(graded, degrees of generators: [ 0, 0, 0 ])
gap> H := InternalHomOnObjects( N, N );
<A graded left module presented by 2 relations for 3 generators>
gap> M := LeftPresentationWithDegrees( mat );
<A graded left module presented by 3 relations for 3 generators>
gap> IsWellDefined( M );
true
gap> 1 * S;
<The graded free left module of rank 1 on a free generator>
gap> 0 * S;
<A graded zero left module>
gap> ZeroObject( A );
<A graded zero left module>
gap> 0 * S = ZeroObject( A );
true
gap> Ms := DualOnObjects( M );
<A graded free left module of rank 1 on a free generator>
gap> Display( Ms );
Q[x,y,z] (with weights [ 1, 1, 1 ])^(1 x 1)

(graded, degree of generator: 1)
gap> pi := EpimorphismFromSomeProjectiveObject( M );
<A homomorphism of left graded modules>
gap> IsWellDefined( pi );
true
gap> Display( pi );
1,0,0,
0,1,0,
0,0,1
(over a graded ring)

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

(degrees of generators of target: [ 0, 0, 0 ])

1.5 Supported CAP operations

1.5-1 Category of of finitely presented graded modules

The following CAP operations are supported:

1.6 GAP categories

1.6-1 IsCategoryOfHomalgGradedModules
‣ IsCategoryOfHomalgGradedModules( arg )( filter )

Returns: true or false

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