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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