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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 an order.



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


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


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



    ModuleBasis: BoolElt                Default: true



Return a homomorphism from M into 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) : ModOrd, ModOrd -> ModOrd, Map



The module of homomorphisms between M and 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 order as M and N.



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



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



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





The map giving the morphism from M to 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 ModOrd_hom (H65E3)

This example demonstrates the use of homomorphisms between modules over
orders. Let M and V be as above.





> 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.2 + 1), 2*(M.2 + 1), M.2 + 1, 
> 0] >;
> h(Mod!(4*V.1));
([12, 0] [8, 0] [4, 0] [0, 0])
> h(Mod!V![0, 1, 0]);
([3, 3] [2, 2] [1, 1] [0, 0])
> I := Image(h);
> I;
free module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[4, 0], [1, 1]> * ( 3/1 2/1 1/1 0 ) 
> K := Kernel(h);
> K;
module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[-1, 0]> * ( -1/1 - $.1 1/1 0 ) 
> IsSubmodule(K, Mod);
true Mapping from: ModOrd: K to ModOrd: Mod
> Hom(Mod, N);
module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
<[1, 0] / 4> car <[1, 0]>
Mapping from: module over Maximal Equation Order with defining polynomial x^2 + 
    5 over Z
<[1, 0] / 4> car <[1, 0]> to Power Structure of Map given by a rule [no inverse]
> H, m := $1;
> m(H![5, 20]);
Mapping from: ModOrd: Mod to ModOrd: N





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