A homalg map is a data structures for maps (module homomorphisms) between finitely generated modules. Each map in homalg knows its source (--> Source
(10.4-1)) and its target (--> Range
(10.4-2)). A map is represented by a homalg matrix relative to the current set of generators of the source resp. target homalg module. As with modules (--> Chapter 9), maps in homalg are realized in an intrinsic manner: If the presentations of the source or/and target module are altered after the map was constructed, a new adapted representation matrix of the map is automatically computed whenever needed. For this the internal transition matrices of the modules are used. homalg uses the so-called associative convention for maps. This means that maps of left modules are applied from the right, whereas maps of right modules from the left.
> IsHomalgMap ( phi ) | ( category ) |
Returns: true
or false
The GAP category of homalg maps.
(It is a subcategory of the GAP category IsHomalgMorphism
.)
> IsHomalgSelfMap ( phi ) | ( category ) |
Returns: true
or false
The GAP category of homalg self-maps.
(It is a subcategory of the GAP categories IsHomalgMap
and IsHomalgEndomorphism
.)
> IsMapOfFinitelyGeneratedModulesRep ( phi ) | ( representation ) |
Returns: true
or false
The GAP representation of maps between finitley generated homalg modules.
(It is a representation of the GAP category IsHomalgChainMap
(12.1-1), which is a subrepresentation of the GAP representation IsMorphismOfFinitelyGeneratedModulesRep
.)
> HomalgMap ( mat, M, N ) | ( function ) |
> HomalgMap ( mat[, string] ) | ( function ) |
Returns: a homalg map
This constructor returns a map (homomorphism) of finitely presented modules. It is represented by the homalg matrix mat relative to the current set of generators of the source homalg module M and target module N (--> 9.2). Unless the source module is free and given on free generators the returned map will cautiously be indicated using parenthesis: "homomorphism". To verify if the result is indeed a well defined map use IsMorphism
(10.3-1). If the presentations of the source or/and target module are altered after the map was constructed, a new adapted representation matrix of the map is automatically computed whenever needed. For this the internal transition matrices of the modules are used. If source and target are identical objects, and only then, the map is created as a selfmap (endomorphism). homalg uses the so-called associative convention for maps. This means that maps of left modules are applied from the right, whereas maps of right modules from the left.
gap> ZZ := HomalgRingOfIntegers( );; gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, ZZ ); <A homalg internal 2 by 3 matrix> gap> M := LeftPresentation( M ); <A non-torsion left module presented by 2 relations for 3 generators> gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, ZZ ); <A homalg internal 2 by 4 matrix> gap> N := LeftPresentation( N ); <A non-torsion left module presented by 2 relations for 4 generators> gap> mat := HomalgMatrix( "[ \ > 1, 0, -2, -4, \ > 0, 1, 4, 7, \ > 1, 0, -2, -4 \ > ]", 3, 4, ZZ );; <A homalg internal 3 by 4 matrix> gap> phi := HomalgMap( mat, M, N ); <A "homomorphism" of left modules> gap> IsMorphism( phi ); true gap> phi; <A homomorphism of left modules> gap> Display( phi ); [ [ 1, 0, -2, -4 ], [ 0, 1, 4, 7 ], [ 1, 0, -2, -4 ] ] the map is currently represented by the above 3 x 4 matrix gap> ByASmallerPresentation( M ); <A rank 1 left module presented by 1 relation for 2 generators> gap> Display( last ); Z/< 3 > + Z^(1 x 1) gap> Display( phi ); [ [ 2, 1, 0, -1 ], [ 1, 0, -2, -4 ] ] the map is currently represented by the above 2 x 4 matrix gap> ByASmallerPresentation( N ); <A rank 2 left module presented by 1 relation for 3 generators> gap> Display( N ); Z/< 4 > + Z^(1 x 2) gap> Display( phi ); [ [ -8, 0, 0 ], [ -3, -1, -2 ] ] the map is currently represented by the above 2 x 3 matrix gap> ByASmallerPresentation( phi ); <A homomorphism of left modules> gap> Display( phi ); [ [ 0, 0, 0 ], [ 1, -1, -2 ] ] the map is currently represented by the above 2 x 3 matrix |
To construct a map with source being a not yet specified free module
gap> N; <A rank 2 left module presented by 1 relation for 3 generators> gap> SetPositionOfTheDefaultSetOfGenerators( N, 1 ); gap> N; <A rank 2 left module presented by 2 relations for 4 generators> gap> psi := HomalgMap( mat, "free", N ); <A homomorphism of left modules> gap> Source( psi ); <A free left module of rank 3 on free generators> |
To construct a map between not yet specified free left modules
gap> chi := HomalgMap( mat ); ## or chi := HomalgMap( mat, "l" ); <A homomorphism of left modules> gap> Source( chi ); <A free left module of rank 3 on free generators> gap> Range( chi ); <A free left module of rank 4 on free generators> |
To construct a map between not yet specified free right modules
gap> kappa := HomalgMap( mat, "r" ); <A homomorphism of right modules> gap> Source( kappa ); <A free right module of rank 4 on free generators> gap> Range( kappa ); <A free right module of rank 3 on free generators> |
> HomalgZeroMap ( M, N ) | ( function ) |
Returns: a homalg map
The constructor returns the zero map between the source homalg module M and the target homalg module N.
gap> ZZ := HomalgRingOfIntegers( );; gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, ZZ ); <A homalg internal 2 by 3 matrix> gap> M := LeftPresentation( M ); <A non-torsion left module presented by 2 relations for 3 generators> gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, ZZ ); <A homalg internal 2 by 4 matrix> gap> N := LeftPresentation( N ); <A non-torsion left module presented by 2 relations for 4 generators> gap> HomalgZeroMap( M, N ); <The zero morphism of left modules> |
> HomalgIdentityMap ( M, N ) | ( function ) |
Returns: a homalg map
The constructor returns the identity map of the homalg module M.
gap> ZZ := HomalgRingOfIntegers( );; gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, ZZ ); <A homalg internal 2 by 3 matrix> gap> M := LeftPresentation( M ); <A non-torsion left module presented by 2 relations for 3 generators> gap> HomalgIdentityMap( M ); <The identity morphism of a left module> |
> IsMorphism ( phi ) | ( property ) |
Returns: true
or false
Check if phi is a well-defined map, i.e. independent of all involved presentations.
> IsGeneralizedMorphism ( phi ) | ( property ) |
Returns: true
or false
Check if phi is a generalized morphism.
> IsGeneralizedEpimorphism ( phi ) | ( property ) |
Returns: true
or false
Check if phi is a generalized epimorphism.
> IsGeneralizedMonomorphism ( phi ) | ( property ) |
Returns: true
or false
Check if phi is a generalized monomorphism.
> IsGeneralizedIsomorphism ( phi ) | ( property ) |
Returns: true
or false
Check if phi is a generalized isomorphism.
> IsIdentityMorphism ( phi ) | ( property ) |
Returns: true
or false
Check if the homalg map phi is the identity morphism.
> IsMonomorphism ( phi ) | ( property ) |
Returns: true
or false
Check if the homalg map phi is a monomorphism.
> IsEpimorphism ( phi ) | ( property ) |
Returns: true
or false
Check if the homalg map phi is an epimorphism.
> IsSplitMonomorphism ( phi ) | ( property ) |
Returns: true
or false
Check if the homalg map phi is a split monomorphism.
> IsSplitEpimorphism ( phi ) | ( property ) |
Returns: true
or false
Check if the homalg map phi is a split epimorphism.
> IsIsomorphism ( phi ) | ( property ) |
Returns: true
or false
Check if the homalg map phi is an isomorphism.
> IsAutomorphism ( phi ) | ( property ) |
Returns: true
or false
Check if the homalg map phi is an automorphism.
> Source ( phi ) | ( attribute ) |
Returns: a homalg module
The source of the homalg map phi.
> Range ( phi ) | ( attribute ) |
Returns: a homalg module
The target (range) of the homalg map phi.
> DegreeOfMorphism ( phi ) | ( attribute ) |
Returns: an integer
The degree of the morphism phi of graded modules.
(no method installed)
> CokernelEpi ( phi ) | ( attribute ) |
Returns: a homalg map
The natural epimorphism from the Range
(phi) onto the Cokernel
(phi) (cf. Cokernel
(16.4-2)).
> CokernelNaturalGeneralizedIsomorphism ( phi ) | ( attribute ) |
Returns: a homalg map
The natural generalized isomorphism from the Cokernel
(phi) onto the Range
(phi) (cf. Cokernel
(16.4-2)).
> KernelSubmodule ( phi ) | ( attribute ) |
Returns: a homalg submodule
This constructor returns the finitely generated kernel of the homalg map phi as a submodule of the homalg module Source
(phi) with generators given by the syzygies of phi.
> KernelEmb ( phi ) | ( attribute ) |
Returns: a homalg map
The natural embedding of the Kernel
(phi) into the Source
(phi) (cf. Kernel
(16.4-6)).
> ImageSubmodule ( phi ) | ( attribute ) |
Returns: a homalg submodule
This constructor returns the finitely generated image of the homalg map phi as a submodule of the homalg module Range
(phi) with generators given by phi applied to the generators of its source module.
> ImageModuleEmb ( phi ) | ( attribute ) |
Returns: a homalg map
The natural embedding of the ImageModule
(phi) into the Range
(phi) (cf. ImageModule
(16.4-4)).
> ImageModuleEpi ( phi ) | ( attribute ) |
Returns: a homalg map
The natural epimorphism from the Source
(phi) onto the ImageModule
(phi) (cf. ImageModule
(16.4-4)).
> MorphismAidMap ( phi ) | ( attribute ) |
Returns: a homalg map
The morphism aid map of a true generalized map.
(no method installed)
> HomalgRing ( phi ) | ( operation ) |
Returns: a homalg ring
The homalg ring of the homalg map phi.
gap> ZZ := HomalgRingOfIntegers( ); <A homalg internal ring> gap> phi := HomalgIdentityMap( 2 * ZZ ); <The identity morphism of a left module> gap> R := HomalgRing( phi ); <A homalg internal ring> gap> IsIdenticalObj( R, ZZ ); true |
> ByASmallerPresentation ( phi ) | ( method ) |
Returns: a homalg map
See ByASmallerPresentation
(9.5-2) on modules.
InstallMethod( ByASmallerPresentation, "for homalg maps", [ IsMapOfFinitelyGeneratedModulesRep ], function( phi ) ByASmallerPresentation( Source( phi ) ); ByASmallerPresentation( Range( phi ) ); DecideZero( phi ); return phi; end ); |
This method performs side effects on its argument phi and returns it.
gap> ZZ := HomalgRingOfIntegers( );; gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, ZZ ); <A homalg internal 2 by 3 matrix> gap> M := LeftPresentation( M ); <A non-torsion left module presented by 2 relations for 3 generators> gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, ZZ ); <A homalg internal 2 by 4 matrix> gap> N := LeftPresentation( N ); <A non-torsion left module presented by 2 relations for 4 generators> gap> mat := HomalgMatrix( "[ \ > 1, 0, -2, -4, \ > 0, 1, 4, 7, \ > 1, 0, -2, -4 \ > ]", 3, 4, ZZ );; <A homalg internal 3 by 4 matrix> gap> phi := HomalgMap( mat, M, N ); <A "homomorphism" of left modules> gap> IsMorphism( phi ); true gap> phi; <A homomorphism of left modules> gap> Display( phi ); [ [ 1, 0, -2, -4 ], [ 0, 1, 4, 7 ], [ 1, 0, -2, -4 ] ] the map is currently represented by the above 3 x 4 matrix gap> ByASmallerPresentation( phi ); <A homomorphism of left modules> gap> Display( phi ); [ [ 0, 0, 0 ], [ 1, -1, -2 ] ] the map is currently represented by the above 2 x 3 matrix gap> M; <A rank 1 left module presented by 1 relation for 2 generators> gap> Display( M ); Z/< 3 > + Z^(1 x 1) gap> N; <A rank 2 left module presented by 1 relation for 3 generators> gap> Display( N ); Z/< 4 > + Z^(1 x 2) |
> PreInverse ( phi ) | ( operation ) |
Returns: a homalg map, false
, or fail
Compute a pre-inverse of the morphism phi in case one exists. For a pre-inverse to exist phi must be an epimorphism. For commutative rings homalg has an algorithm installed which decides the existence and returns a pre-inverse in case one exists. If a pre-inverse does not exist then false
is returned. The algorithm finds a particular solution of a two-side inhomogeneous linear system over R :=HomalgRing
( phi ). For noncommutative rings a heuristic method is installed. If it finds a pre-inverse it returns it, otherwise it returns fail
(--> Principal limitation). The operation PreInverse
is used to install a method for the property IsSplitEpimorphism
(10.3-10).
PreInverse
checks if it can decide the projectivity of Range
( phi ). To decide the projectivity of a module M over a commutative ring you can use
IsSplitEpimorphism
( FreeHullEpi
( M ) );
Of course you can use IsProjective
( M ) which triggers other methods.
generated by GAPDoc2HTML