GAP (ModulePresentationsForCAP) - Chapter 1: Module Presentations

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1 Module Presentations
1.1 Functors
1.1-1 FunctorStandardModuleLeft
1.1-2 FunctorStandardModuleRight
1.1-3 FunctorGetRidOfZeroGeneratorsLeft
1.1-4 FunctorGetRidOfZeroGeneratorsRight
1.1-5 FunctorLessGeneratorsLeft
1.1-6 FunctorLessGeneratorsRight
1.1-7 FunctorDualLeft
1.1-8 FunctorDualRight
1.1-9 FunctorDoubleDualLeft
1.1-10 FunctorDoubleDualRight
1.2 GAP Categories
1.2-1 IsLeftOrRightPresentationMorphism
1.2-2 IsLeftPresentationMorphism
1.2-3 IsRightPresentationMorphism
1.2-4 IsLeftOrRightPresentation
1.2-5 IsLeftPresentation
1.2-6 IsRightPresentation
1.3 Constructors
1.3-1 PresentationMorphism
1.3-2 AsMorphismBetweenFreeLeftPresentations
1.3-3 AsMorphismBetweenFreeRightPresentations
1.3-4 AsLeftPresentation
1.3-5 AsRightPresentation
1.3-6 AsLeftOrRightPresentation
1.3-7 FreeLeftPresentation
1.3-8 FreeRightPresentation
1.3-9 UnderlyingMatrix
1.3-10 UnderlyingHomalgRing
1.3-11 Annihilator
1.3-12 LeftPresentations
1.3-13 RightPresentations
1.4 Attributes
1.4-1 UnderlyingHomalgRing
1.4-2 UnderlyingMatrix
1.5 Non-Categorical Operations
1.5-1 StandardGeneratorMorphism
1.5-2 CoverByFreeModule
1.6 Natural Transformations
1.6-1 NaturalIsomorphismFromIdentityToStandardModuleLeft
1.6-2 NaturalIsomorphismFromIdentityToStandardModuleRight
1.6-3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft
1.6-4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight
1.6-5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft
1.6-6 NaturalIsomorphismFromIdentityToLessGeneratorsRight
1.6-7 NaturalTransformationFromIdentityToDoubleDualLeft
1.6-8 NaturalTransformationFromIdentityToDoubleDualRight

### 1 Module Presentations

#### 1.1 Functors

##### 1.1-1 FunctorStandardModuleLeft

`‣ FunctorStandardModuleLeft`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-2 FunctorStandardModuleRight

`‣ FunctorStandardModuleRight`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-3 FunctorGetRidOfZeroGeneratorsLeft

`‣ FunctorGetRidOfZeroGeneratorsLeft`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-4 FunctorGetRidOfZeroGeneratorsRight

`‣ FunctorGetRidOfZeroGeneratorsRight`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-5 FunctorLessGeneratorsLeft

`‣ FunctorLessGeneratorsLeft`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-6 FunctorLessGeneratorsRight

`‣ FunctorLessGeneratorsRight`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-7 FunctorDualLeft

`‣ FunctorDualLeft`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-8 FunctorDualRight

`‣ FunctorDualRight`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-9 FunctorDoubleDualLeft

`‣ FunctorDoubleDualLeft`( R ) | ( attribute )

Returns: a functor

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

##### 1.1-10 FunctorDoubleDualRight

`‣ FunctorDoubleDualRight`( R ) | ( attribute )

Returns: a functor

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

#### 1.2 GAP Categories

##### 1.2-1 IsLeftOrRightPresentationMorphism

`‣ IsLeftOrRightPresentationMorphism`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of morphisms in the category of left or right presentations.

##### 1.2-2 IsLeftPresentationMorphism

`‣ IsLeftPresentationMorphism`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of morphisms in the category of left presentations.

##### 1.2-3 IsRightPresentationMorphism

`‣ IsRightPresentationMorphism`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of morphisms in the category of right presentations.

##### 1.2-4 IsLeftOrRightPresentation

`‣ IsLeftOrRightPresentation`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of objects in the category of left presentations or right presentations.

##### 1.2-5 IsLeftPresentation

`‣ IsLeftPresentation`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of objects in the category of left presentations.

##### 1.2-6 IsRightPresentation

`‣ IsRightPresentation`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of objects in the category of right presentations.

#### 1.3 Constructors

##### 1.3-1 PresentationMorphism

`‣ PresentationMorphism`( A, M, B ) | ( operation )

Returns: a morphism in \(\mathrm{Hom}(A,B)\)

The arguments are an object \(A\), a homalg matrix \(M\), and another object \(B\). \(A\) and \(B\) shall either both be objects in the category of left presentations or both be objects in the category of right presentations. The output is a morphism \(A \rightarrow B\) in the the category of left or right presentations whose underlying matrix is given by \(M\).

##### 1.3-2 AsMorphismBetweenFreeLeftPresentations

`‣ AsMorphismBetweenFreeLeftPresentations`( m ) | ( attribute )

Returns: a morphism in \(\mathrm{Hom}(F^r,F^c)\)

The argument is a homalg matrix \(m\). The output is a morphism \(F^r \rightarrow F^c\) in the the category of left presentations whose underlying matrix is given by \(m\), where \(F^r\) and \(F^c\) are free left presentations of ranks given by the number of rows and columns of \(m\).

##### 1.3-3 AsMorphismBetweenFreeRightPresentations

`‣ AsMorphismBetweenFreeRightPresentations`( m ) | ( attribute )

Returns: a morphism in \(\mathrm{Hom}(F^c,F^r)\)

The argument is a homalg matrix \(m\). The output is a morphism \(F^c \rightarrow F^r\) in the the category of right presentations whose underlying matrix is given by \(m\), where \(F^r\) and \(F^c\) are free right presentations of ranks given by the number of rows and columns of \(m\).

##### 1.3-4 AsLeftPresentation

`‣ AsLeftPresentation`( M ) | ( operation )

Returns: an object

The argument is a homalg matrix \(M\) over a ring \(R\). The output is an object in the category of left presentations over \(R\). This object has \(M\) as its underlying matrix.

##### 1.3-5 AsRightPresentation

`‣ AsRightPresentation`( M ) | ( operation )

Returns: an object

The argument is a homalg matrix \(M\) over a ring \(R\). The output is an object in the category of right presentations over \(R\). This object has \(M\) as its underlying matrix.

##### 1.3-6 AsLeftOrRightPresentation

`‣ AsLeftOrRightPresentation`( M, l ) | ( function )

Returns: an object

The arguments are a homalg matrix \(M\) and a boolean \(l\). If \(l\) is `true`, the output is an object in the category of left presentations. If \(l\) is `false`, the output is an object in the category of right presentations. In both cases, the underlying matrix of the result is \(M\).

##### 1.3-7 FreeLeftPresentation

`‣ FreeLeftPresentation`( r, R ) | ( operation )

Returns: an object

The arguments are a non-negative integer \(r\) and a homalg ring \(R\). The output is an object in the category of left presentations over \(R\). It is represented by the \(0 \times r\) matrix and thus it is free of rank \(r\).

##### 1.3-8 FreeRightPresentation

`‣ FreeRightPresentation`( r, R ) | ( operation )

Returns: an object

The arguments are a non-negative integer \(r\) and a homalg ring \(R\). The output is an object in the category of right presentations over \(R\). It is represented by the \(r \times 0\) matrix and thus it is free of rank \(r\).

##### 1.3-9 UnderlyingMatrix

`‣ UnderlyingMatrix`( A ) | ( attribute )

Returns: a homalg matrix

The argument is an object \(A\) in the category of left or right presentations over a homalg ring \(R\). The output is the underlying matrix which presents \(A\).

##### 1.3-10 UnderlyingHomalgRing

`‣ UnderlyingHomalgRing`( A ) | ( attribute )

Returns: a homalg ring

The argument is an object \(A\) in the category of left or right presentations over a homalg ring \(R\). The output is \(R\).

##### 1.3-11 Annihilator

`‣ Annihilator`( A ) | ( attribute )

Returns: a morphism in \(\mathrm{Hom}(I, F)\)

The argument is an object \(A\) in the category of left or right presentations. The output is the embedding of the annihilator \(I\) of \(A\) into the free module \(F\) of rank \(1\). In particular, the annihilator itself is seen as a left or right presentation.

##### 1.3-12 LeftPresentations

`‣ LeftPresentations`( R ) | ( attribute )

Returns: a category

The argument is a homalg ring \(R\). The output is the category of free left presentations over \(R\).

##### 1.3-13 RightPresentations

`‣ RightPresentations`( R ) | ( attribute )

Returns: a category

The argument is a homalg ring \(R\). The output is the category of free right presentations over \(R\).

#### 1.4 Attributes

##### 1.4-1 UnderlyingHomalgRing

`‣ UnderlyingHomalgRing`( R ) | ( attribute )

Returns: a homalg ring

The argument is a morphism \(\alpha\) in the category of left or right presentations over a homalg ring \(R\). The output is \(R\).

##### 1.4-2 UnderlyingMatrix

`‣ UnderlyingMatrix`( alpha ) | ( attribute )

Returns: a homalg matrix

The argument is a morphism \(\alpha\) in the category of left or right presentations. The output is its underlying homalg matrix.

#### 1.5 Non-Categorical Operations

##### 1.5-1 StandardGeneratorMorphism

`‣ StandardGeneratorMorphism`( A, i ) | ( operation )

Returns: a morphism in \(\mathrm{Hom}(F, A)\)

The argument is an object \(A\) in the category of left or right presentations over a homalg ring \(R\) with underlying matrix \(M\) and an integer \(i\). The output is a morphism \(F \rightarrow A\) given by the \(i\)-th row or column of \(M\), where \(F\) is a free left or right presentation of rank \(1\).

##### 1.5-2 CoverByFreeModule

`‣ CoverByFreeModule`( A ) | ( attribute )

Returns: a morphism in \(\mathrm{Hom}(F,A)\)

The argument is an object \(A\) in the category of left or right presentations. The output is a morphism from a free module \(F\) to \(A\), which maps the standard generators of the free module to the generators of \(A\).

#### 1.6 Natural Transformations

##### 1.6-1 NaturalIsomorphismFromIdentityToStandardModuleLeft

`‣ NaturalIsomorphismFromIdentityToStandardModuleLeft`( R ) | ( attribute )

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

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

##### 1.6-2 NaturalIsomorphismFromIdentityToStandardModuleRight

`‣ NaturalIsomorphismFromIdentityToStandardModuleRight`( R ) | ( attribute )

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

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

##### 1.6-3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft

`‣ NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft`( R ) | ( attribute )

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

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

##### 1.6-4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight

`‣ NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight`( R ) | ( attribute )

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

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

##### 1.6-5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft

`‣ NaturalIsomorphismFromIdentityToLessGeneratorsLeft`( R ) | ( attribute )

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

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

##### 1.6-6 NaturalIsomorphismFromIdentityToLessGeneratorsRight

`‣ NaturalIsomorphismFromIdentityToLessGeneratorsRight`( R ) | ( attribute )

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

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

##### 1.6-7 NaturalTransformationFromIdentityToDoubleDualLeft

`‣ NaturalTransformationFromIdentityToDoubleDualLeft`( R ) | ( attribute )

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

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

##### 1.6-8 NaturalTransformationFromIdentityToDoubleDualRight

`‣ NaturalTransformationFromIdentityToDoubleDualRight`( R ) | ( attribute )

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

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

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