ModulePresentationMorphism.gd

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##                                       ModulePresentationsForCAP package
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##  Copyright 2014, Sebastian Gutsche, TU Kaiserslautern
##                  Sebastian Posur,   RWTH Aachen
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#! @Chapter Module Presentations
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#! @Section GAP Categories
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#! @Description
#! The GAP category of morphisms in the category
#! of left or right presentations.
#! @Arguments object
DeclareCategory( "IsLeftOrRightPresentationMorphism",
                 IsCapCategoryMorphism );

#! @Description
#! The GAP category of morphisms in the category
#! of left presentations.
#! @Arguments object
DeclareCategory( "IsLeftPresentationMorphism",
                 IsLeftOrRightPresentationMorphism );

#! @Description
#! The GAP category of morphisms in the category
#! of right presentations.
#! @Arguments object
DeclareCategory( "IsRightPresentationMorphism",
                 IsLeftOrRightPresentationMorphism );

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#! @Section Constructors
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#! @Description
#! 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 $M$.
#! @Returns a morphism in $\mathrm{Hom}(A,B)$
#! @Arguments A, M, B
DeclareOperation( "PresentationMorphism",
                  [ IsLeftOrRightPresentation, IsHomalgMatrix, IsLeftOrRightPresentation ] );

#! @Description
#! 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 $m$.
#! @Returns a morphism in $\mathrm{Hom}(F^r,F^c)$
#! @Arguments m
DeclareAttribute( "AsMorphismBetweenFreeLeftPresentations",
                  IsHomalgMatrix );

#! @Description
#! 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 $m$.
#! @Returns a morphism in $\mathrm{Hom}(F^c,F^r)$
#! @Arguments m
DeclareAttribute( "AsMorphismBetweenFreeRightPresentations",
                  IsHomalgMatrix );

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#! @Section Attributes
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#! @Description
#! The argument is a morphism $\alpha$ in the category
#! of left or right presentations over a homalg ring $R$.
#! The output is $R$.
#! @Returns a homalg ring
#! @Arguments R
DeclareAttribute( "UnderlyingHomalgRing",
                  IsLeftOrRightPresentationMorphism );

#! @Description
#! The argument is a morphism $\alpha$ in the category
#! of left or right presentations.
#! The output is its underlying homalg matrix.
#! @Returns a homalg matrix
#! @Arguments alpha
DeclareAttribute( "UnderlyingMatrix",
                  IsLeftOrRightPresentationMorphism );

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## Arithmetics
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DeclareOperation( "\*",
                  [ IsRingElement, IsLeftPresentationMorphism ] );

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DeclareOperation( "\*",
                  [ IsRightPresentationMorphism, IsRingElement ] );


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#! @Section Non-Categorical Operations
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#! @Description
#! 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 $1$.
#! @Returns a morphism in $\mathrm{Hom}(F, A)$
#! @Arguments A, i
DeclareOperation( "StandardGeneratorMorphism",
                  [ IsLeftOrRightPresentation, IsInt ] );

#! @Description
#! 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 $A$.
#! @Returns a morphism in $\mathrm{Hom}(F,A)$
#! @Arguments A
DeclareAttribute( "CoverByFreeModule",
                  IsLeftOrRightPresentation );