‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ IsLeftOrRightPresentationMorphism( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the category of left or right presentations.
‣ IsLeftPresentationMorphism( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the category of left presentations.
‣ IsRightPresentationMorphism( object ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in the category of right presentations.
‣ IsLeftOrRightPresentation( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the category of left presentations or right presentations.
‣ IsLeftPresentation( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the category of left presentations.
‣ IsRightPresentation( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the category of right presentations.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ LeftPresentations( R ) | ( attribute ) |
Returns: a category
The argument is a homalg ring R. The output is the category of free left presentations over R.
‣ RightPresentations( R ) | ( attribute ) |
Returns: a category
The argument is a homalg ring R. The output is the category of free right presentations over R.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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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