Goto Chapter: Top 1 2 Ind

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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