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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 ModDed_hom (H55E6)

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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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013