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.ModuleHomomorphismMethod
ModuleHomomorphism(M1::FPModule{T},
                      M2::FPModule{T}, m::MatElem{T}) where T <: RingElement

Create the homomorphism $f : M_1 \to M_2$ represented by the matrix $m$.

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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.

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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 $f$

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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 $f$

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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.

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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.

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

g = inv(f)