# Module Homomorphisms

Abstract Algebra provides homomorphisms of finitely presented modules.

## Generic module homomorphism types

AbstractAlgebra defines two module homomorphism types, namely Generic.ModuleHomomorphism and Generic.ModuleIsomorphism. Functionality for these is implemented in src/generic/ModuleHomomorphism.jl.

## Abstract types

The Generic.ModuleHomomorphism and Generic.ModuleIsomorphism types inherit from Map(FPModuleHomomorphism).

## Generic functionality

The following generic functionality is provided for module homomorphisms.

### Constructors

Homomorphisms of AbstractAlgebra modules, $f : R^s \to R^t$, can be represented by $s\times t$ matrices over $R$.

AbstractAlgebra.ModuleIsomorphismMethod
ModuleIsomorphism(M1::FPModule{T}, M2::FPModule{T}, M::MatElem{T},
minv::MatElem{T}) where T <: RingElement

Create the isomorphism $f : M_1 \to M_2$ represented by the matrix $M$. The inverse morphism is automatically computed.

source

Examples

julia> M = FreeModule(ZZ, 2)
Free module of rank 2 over Integers

julia> f = ModuleHomomorphism(M, M, matrix(ZZ, 2, 2, [1, 2, 3, 4]))
Module homomorphism with
Domain: Free module of rank 2 over Integers
Codomain: Free module of rank 2 over Integers

julia> m = M([ZZ(1), ZZ(2)])
(1, 2)

julia> f(m)
(7, 10)


They can also be created by giving images (in the codomain) of the generators of the domain:

ModuleHomomorphism(M1::FPModule{T}, M2::FPModule{T}, v::Vector{<:FPModuleElem{T}}) where T <: RingElement

### Kernels

AbstractAlgebra.kernelMethod
kernel(f::ModuleHomomorphism{T}) where T <: RingElement

Return a pair K, g consisting of the kernel object $K$ of the given module homomorphism $f$ (as a submodule of its domain) and the canonical injection from the kernel into the domain of $f$

source

Examples

julia> M = FreeModule(ZZ, 3)
Free module of rank 3 over Integers

julia> m = M([ZZ(1), ZZ(2), ZZ(3)])
(1, 2, 3)

julia> S, f = sub(M, [m])
(Submodule over Integers with 1 generator and no relations
, Module homomorphism with
Domain: Submodule over Integers with 1 generator and no relations

Codomain: Free module of rank 3 over Integers)

julia> Q, g = quo(M, S)
(Quotient module over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Free module of rank 3 over Integers
Codomain: Quotient module over Integers with 2 generators and no relations
)

julia> kernel(g)
(Submodule over Integers with 1 generator and no relations
, Module homomorphism with
Domain: Submodule over Integers with 1 generator and no relations

Codomain: Free module of rank 3 over Integers)


### Images

AbstractAlgebra.imageMethod
image(f::Map(FPModuleHomomorphism))

Return a pair I, g consisting of the image object $I$ of the given module homomorphism $f$ (as a submodule of its codomain) and the canonical injection from the image into the codomain of $f$

source
M = FreeModule(ZZ, 3)

m = M([ZZ(1), ZZ(2), ZZ(3)])

S, f = sub(M, [m])
Q, g = quo(M, S)
K, k = kernel(g)

image(compose(k, g))

### Preimages

AbstractAlgebra.preimageMethod
preimage(f::Map(FPModuleHomomorphism),
v::FPModuleElem{T}) where T <: RingElement

Return a preimage of $v$ under the homomorphism $f$, i.e. an element of the domain of $f$ that maps to $v$ under $f$. Note that this has no special mathematical properties. It is an element of the set theoretical preimage of the map $f$ as a map of sets, if one exists. The preimage is neither unique nor chosen in a canonical way in general. When no such element exists, an exception is raised.

source
M = FreeModule(ZZ, 3)

m = M([ZZ(1), ZZ(2), ZZ(3)])

S, f = sub(M, [m])
Q, g = quo(M, S)

m = rand(M, -10:10)
n = g(m)

p = preimage(g, n)

### Inverses

Module isomorphisms can be cheaply inverted.

Base.invMethod
Base.inv(f::Map(ModuleIsomorphism))

Return the inverse map of the given module isomorphism. This is computed cheaply.

source
M = FreeModule(ZZ, 2)
N = matrix(ZZ, 2, 2, BigInt[1, 0, 0, 1])
f = ModuleIsomorphism(M, M, N)

g = inv(f)