Free Modules

Free Modules

In this section, the expression free module refers to a free module of finite rank over a ring of type MPolyRing, MPolyQuo, MPolyLocalizedRing, or MPolyQuoLocalizedRing. More concretely, given a ring $R$ of one of these types, the free $R$-modules considered are of type $R^p$, where we think of $R^p$ as a free module with a given basis, namely the basis of standard unit vectors. Accordingly, elements of free modules are represented by coordinate vectors, and homomorphisms between free modules by matrices.

Note

By convention, vectors are row vectors, and matrices operate by multiplication on the right.

Types

All OSCAR types for the modules considered here belong to the abstract type ModuleFP{T}, where T is the element type of the underlying ring. The free modules belong to the abstract subtype AbstractFreeMod{T} <: ModuleFP{T}, they are modelled as objects of the concrete type FreeMod{T} <: AbstractFreeMod{T}.

Note

Canonical maps such us the canonical projection onto a quotient module arise in many constructions in commutative algebra. The FreeMod type is designed so that it allows for the caching of such maps when executing functions. The direct_sum function discussed in this section provides an example.

Constructors

free_module — Function

free_module(R::MPolyRing, p::Int, name::String = "e"; cached::Bool = false)

free_module(R::MPolyQuo, p::Int, name::String = "e"; cached::Bool = false)

free_module(R::MPolyLocalizedRing, p::Int, name::String = "e"; cached::Bool = false)

free_module(R::MPolyQuoLocalizedRing, p::Int, name::String = "e"; cached::Bool = false)

Return the free $R$-module $R^p$, created with its basis of standard unit vectors.

The string name specifies how the basis vectors are printed.

Examples

julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"]);

julia> FR = free_module(R, 2)
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> x*FR[1]
x*e[1]

julia> P = ideal(R, [x, y, z]);

julia> U = complement_of_ideal(P);

julia> RL, _ = Localization(R, U);

julia> FRL = free_module(RL, 2, "f")
Free module of rank 2 over localization of Multivariate Polynomial Ring in x, y, z over Rational Field at the complement of ideal(x, y, z)

julia> RL(x)*FRL[1]
x//1*f[1]

julia> RQ, _ = quo(R, ideal(R, [2*x^2-y^3, 2*x^2-y^5]));

julia> FRQ =  free_module(RQ, 2, "g")
Free module of rank 2 over Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by ideal(2*x^2 - y^3, 2*x^2 - y^5)

julia> RQ(x)*FRQ[1]
x*g[1]

julia> RQL, _ = Localization(RQ, U);

julia> FRQL =  free_module(RQL, 2, "h")
Free module of rank 2 over Localization of Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by ideal(2*x^2 - y^3, 2*x^2 - y^5) at the multiplicative set complement of ideal(x, y, z)

julia> RQL(x)*FRQL[1]
x//1*h[1]

source

Data Associated to Free Modules

If F is a free R-module, then

base_ring(F) refers to R,
basis(F), gens(F) to the basis vectors of F,
rank(F), ngens(F), dim(F) to the number of these vectors, and
F[i], basis(F, i), gen(F, i) to the i-th such vector.

Examples

julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"]);

julia> F = free_module(R, 3);

julia> basis(F)
3-element Vector{FreeModElem{fmpq_mpoly}}:
 e[1]
 e[2]
 e[3]

julia> rank(F)
3

Elements of Free Modules

All OSCAR types for elements of the modules considered here belong to the abstract type ModuleElemFP{T}, where T is the element type of the underlying ring. The free modules belong to the abstract subtype AbstractFreeModElem{T} <: ModuleFPElem{T}. They are modelled as objects of the concrete type FreeModElem{T} <: AbstractFreeModElem{T} which implements an element $f$ of a free module $F$ as a sparse row, that is, as an object of type SRow{T}. This object specifies the coordinates of $f$ with respect to the basis of standard unit vectors of $F$. To create an element, enter its coordinates as a sparse row or a vector:

(F::FreeMod{T})(c::SRow{T}) where T

(F::FreeMod{T})(c::Vector{T}) where T

Alternatively, directly write the element as a linear combination of basis vectors of $F$:

Examples

julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"]);

julia> F = free_module(R, 3);

julia> f = F(sparse_row(R, [(1,x),(3,y)]))
x*e[1] + y*e[3]

julia> g = F([x, zero(R), y])
x*e[1] + y*e[3]

julia> h = x*F[1] + y*F[3]
x*e[1] + y*e[3]

julia> f == g == h
true

Given an element f of a free module F over a multivariate polynomial ring with element type T,

parent(f) refers to F, and
coordinates(f) to the coordinate vector of f, returned as an object of type SRow{T}.

Examples

julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"]);

julia> F = free_module(R, 3);

julia> f = x*F[1] + y*F[3]
x*e[1] + y*e[3]

julia> parent(f)
Free module of rank 3 over Multivariate Polynomial Ring in x, y over Rational Field

julia> coordinates(f)
Sparse row with positions [1, 3] and values fmpq_mpoly[x, y]

The zero element of a free module is obtained as follows:

zero — Method

zero(F::AbstractFreeMod)

Return the zero element of F.

source

Whether a given element of a free module is zero can be tested as follows:

iszero — Method

iszero(f::AbstractFreeModElem)

Return true if f is zero, false otherwise.

source

Tests on Free Modules

== — Method

==(F::FreeMod, G::FreeMod)

Return true if F and G are equal, false otherwise.

Here, F and G are equal iff their base rings, ranks, and names for printing the basis elements are equal.

source

iszero — Method

iszero(F::AbstractFreeMod)

Return true if F is the zero module, false otherwise.

source

Homomorphisms from Free Modules

All OSCAR types for homomorphisms of the modules considered here belong to the abstract type ModuleFPHom{T1, T2}, where T1 and T2 are the types of domain and codomain respectively. A homomorphism $F\rightarrow M$ from a free module $F$ is determined by specifying the images of the basis vectors of $F$ in $M$. For such homomorphisms, OSCAR provides the concrete type FreeModuleHom{T1, T2} <: ModuleFPHom{T1, T2} as well as the following constructors:

hom — Method

hom(F::FreeMod, M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}) where T

Given a vector V of rank(F) elements of M, return the homomorphism F $\to$ M which sends the i-th basis vector of F to the i-th entry of V.

hom(F::FreeMod, M::ModuleFP{T}, A::MatElem{T}) where T

Given a matrix A with rank(F) rows and ngens(M) columns, return the homomorphism F $\to$ M which sends the i-th basis vector of F to the linear combination $\sum_j A[i,j]*M[j]$ of the generators M[j] of M.

Examples

julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F = free_module(R, 3)
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> G = free_module(R, 2)
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
3-element Vector{FreeModElem{fmpq_mpoly}}:
 y*e[1]
 x*e[1] + y*e[2]
 z*e[2]

julia> a = hom(F, G, V)
Map with following data
Domain:
=======
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field
Codomain:
=========
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> a(F[2])
x*e[1] + y*e[2]

julia> B = R[y 0; x y; 0 z]
[y   0]
[x   y]
[0   z]

julia> b = hom(F, G, B)
Map with following data
Domain:
=======
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field
Codomain:
=========
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> a == b
true

source

Given a homomorphism of type FreeModuleHom, a matrix representing it is recovered by the following function:

matrix — Method

matrix(a::FreeModuleHom)

Given a homomorphism a : F → M of type FreeModuleHom, return a matrix A over base_ring(M) with rank(F) rows and ngens(M) columns such that $a(F[i]) = \sum_j A[i,j]*M[j]$.

Examples

julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> F = free_module(R, 3)
Free module of rank 3 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> G = free_module(R, 2)
Free module of rank 2 over Multivariate Polynomial Ring in x, y, z over Rational Field

julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]];

julia> a = hom(F, G, V);

julia> matrix(a)
[y   0]
[x   y]
[0   z]

source

The domain and codomain of a homomorphism a of type FreeModuleHom can be recovered by entering domain(a) and codomain(a), respectively.