chap1.txt

  
  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.