Homomorphisms between Modules

It is possible to create a homomorphism between two modules, take the image and kernel of such and verify that these are submodules of the codomain and domain respectively. The Hom–module can also be created as a module of a Dedekind domain.

hom<M -> N | T>: ModDed, ModDed, Map -> Map

hom<M -> N | T>: ModDed, ModDed, SeqEnum[ModTupRngElt] -> Map

hom<M -> N | T>: ModDed, ModDed, MtrxSpcElt -> Map

ModuleBasis: BoolElt                    Default: true

Return a homomorphism from the module \(M\) into the module \(N\) as specified by \(T\) from which the images of the generators can be inferred. \(T\) may be a map between the vector spaces of same degree as \(M\) and \(N\), a matrix over the field of fractions or a sequence of vectors. If ModuleBasis is true then the matrix will be taken to be a transformation between the modules and as such will be expected to have size Dimension(M)\(*\)Dimension(N) otherwise it will be interpreted as a transformation between the corresponding vector spaces and will be expected to have size Degree(M)\(*\)Degree(N).

Hom(M, N): ModDed, ModDed -> ModDed, Map

The module of homomorphisms between the module \(M\) and the module \(N\) and the map from the hom–module to the collection of maps from \(M\) to \(N\), (such that given an element of the hom–module a homomorphism from \(M\) to \(N\) is returned). The module is over the same Dedekind domain as \(M\) and \(N\).

IsSubmodule(M, N): ModDed, ModDed -> BoolElt, Map

Return true if \(M\) is a submodule of \(N\) and the map embedding \(M\) into \(N\).

Morphism(M, N): ModDed, ModDed -> Map

The map giving the morphism from the module \(M\) to the module \(N\). Either \(M\) is a submodule of \(N\), in which case the embedding of \(M\) into \(N\) is returned, or \(N\) is a quotient module of \(M\), in which case the natural epimorphism from \(M\) onto \(N\) is returned.

Example: Homomorphisms Between Modules over Dedekind Domains (ex-f6bf14)

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This example demonstrates the use of homomorphisms between modules over Dedekind domains. Let M and V be as above referring to function fields.

> S := [V|[0,1,0], [4,4,0]];
> Mod := Module(S);
> W := KModule(FieldOfFractions(M), 4);
> S := [W|[3, 2, 1, 0]];
> N := Module(S);
> h := hom<Mod -> N | >;
>> h := hom<Mod -> N | >;
           ^
Runtime error in map< ... >: No images given
> h := hom<Mod -> N | V.1, V.2, V.3>;

>> h := hom<Mod -> N | V.1, V.2, V.3>;
           ^
Runtime error in map< ... >: An image for each generator is required
> h := hom<Mod -> N | W![3, 2, 1, 0], W![3*(M!F.1 + 1), 2*(M!F.1 + 1),
> M!F.1 + 1, 0] >;
> h(Mod!(4*V.1));
( 4 )
> h(Mod!V![0, 1, 0]);
( x^2 + 1 )
> I := Image(h);
> I;
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x
over Rational Field
Ideal of M
Generator:
1
> K := Kernel(h);
> K;
Integral Module over Maximal Equation Order of F over Univariate Polynomial Ring
in x over Rational Field
Ideal of M
Generator:
1
> IsSubmodule(K, Mod);
true Mapping from: ModDed: K to ModDed: Mod
> H, m := Hom(Mod, N);
> H; m;
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x over
Rational Field
 generated by: (in echelon form)
Ideal of M
Generator:
1/4 * ( 1 0 )
Ideal of M
Generator:
1 * ( 0 1 )
Mapping from: ModDed: H to Power Structure of Map given by a rule [no inverse]
> m(H![5, 20]);
Mapping from: ModDed: Mod to ModDed: N
using
[5]
[20]

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