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# SPDX-License-Identifier: GPL-2.0-or-later
# ModulePresentationsForCAP: Category R-pres for CAP
#
# Declarations
#
#! @Chapter Module Presentations
#############################
##
#! @Section GAP Categories
##
#############################
#! @Description
#! The GAP category of objects in the category
#! of left presentations or right presentations.
#! @Arguments object
DeclareCategory( "IsLeftOrRightPresentation",
IsCapCategoryObject );
#! @Description
#! The GAP category of objects in the category
#! of left presentations.
#! @Arguments object
DeclareCategory( "IsLeftPresentation",
IsLeftOrRightPresentation );
#! @Description
#! The GAP category of objects in the category
#! of right presentations.
#! @Arguments object
DeclareCategory( "IsRightPresentation",
IsLeftOrRightPresentation );
#############################
##
#! @Section Constructors
##
#############################
#! @Description
#! 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.
#! @Returns an object
#! @Arguments M
DeclareOperation( "AsLeftPresentation",
[ IsHomalgMatrix ] );
#! @Description
#! 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.
#! @Returns an object
#! @Arguments M
DeclareOperation( "AsRightPresentation",
[ IsHomalgMatrix ] );
#! @Description
#! 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 $r$.
#! @Returns an object
#! @Arguments r, R
DeclareOperation( "FreeLeftPresentation",
[ IsInt, IsHomalgRing ] );
#! @Description
#! 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 $r$.
#! @Returns an object
#! @Arguments r, R
DeclareOperation( "FreeRightPresentation",
[ IsInt, IsHomalgRing ] );
#############################
##
## Properties
##
#############################
## TODO
DeclareProperty( "IsFree",
IsCapCategoryMorphism );
#############################
##
## Attributes
##
#############################
#! @Description
#! 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 $A$.
#! @Returns a homalg matrix
#! @Arguments A
DeclareAttribute( "UnderlyingMatrix",
IsLeftOrRightPresentation );
CapJitAddTypeSignature( "UnderlyingMatrix", [ IsLeftOrRightPresentation ], function ( input_types )
Assert( 0, IsHomalgRing( UnderlyingRing( input_types[1].category ) ) );
return rec(
filter := IsHomalgMatrix,
ring := UnderlyingRing( input_types[1].category ),
);
end );
#! @Description
#! The argument is an object $A$ in the category of left or right presentations
#! over a homalg ring $R$.
#! The output is $R$.
#! @Returns a homalg ring
#! @Arguments A
DeclareAttribute( "UnderlyingHomalgRing",
IsLeftOrRightPresentation );
#! @Description
#! 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.
#! @Returns a morphism in $\mathrm{Hom}(I, F)$
#! @Arguments A
DeclareAttribute( "Annihilator",
IsLeftOrRightPresentation );
##############################################
##
## Non-categorical methods
##
##############################################
DeclareOperationWithCache( "INTERNAL_HOM_EMBEDDING_IN_TENSOR_PRODUCT_LEFT",
[ IsLeftOrRightPresentation, IsLeftOrRightPresentation ] );
DeclareOperationWithCache( "INTERNAL_HOM_EMBEDDING_IN_TENSOR_PRODUCT_RIGHT",
[ IsLeftOrRightPresentation, IsLeftOrRightPresentation ] );
[ Dauer der Verarbeitung: 0.32 Sekunden
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