Goto Chapter: Top 1 2 Ind
 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 

1 Module Presentations
 1.1 Functors
 1.2 GAP Categories
 1.3 Constructors
 1.4 Attributes
 1.5 Non-Categorical Operations
 1.6 Natural Transformations

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 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 functor which takes a right presentation as input and computes its standard presentation.

1.1-3 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 it a presentation having less generators.

1.1-4 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 it a presentation having less generators.

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 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.

1.3-3 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.

1.3-4 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.

1.3-5 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.

1.3-6 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.

1.3-7 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.

1.3-8 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.

1.3-9 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.

1.3-10 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 morphism 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 morphism from the identity functor to the right standard module functor.

 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
Goto Chapter: Top 1 2 Ind

generated by GAPDoc2HTML