Free Modules
In this section, the expression free module refers to a free module of finite rank over a ring of type MPolyRing, MPolyQuoRing, MPolyLocRing, or MPolyQuoLocRing. 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.
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. Graded or not, 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}.
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 — Functionfree_module(R::MPolyRing, p::Int, name::VarName = :e; cached::Bool = false)
free_module(R::MPolyQuoRing, p::Int, name::VarName = :e; cached::Bool = false)
free_module(R::MPolyLocRing, p::Int, name::VarName = :e; cached::Bool = false)
free_module(R::MPolyQuoLocRing, p::Int, name::VarName = :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) = polynomial_ring(QQ, ["x", "y", "z"]);
julia> FR = free_module(R, 2)
Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
julia> x*FR[1]
x*e[1]
julia> P = ideal(R, [x, y, z]);
julia> U = complement_of_prime_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 3 variables over QQ at complement of prime ideal
julia> RL(x)*FRL[1]
x*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 RQ
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 at complement of prime ideal
julia> RQL(x)*FRQL[1]
x*h[1]Over graded multivariate polynomial rings and their quotients, there are two basic ways of creating graded free modules: While the grade function allows one to create a graded free module by assigning a grading to a free module already constructed, the graded_free_module function is meant to create a graded free module all at once.
grade — Methodgrade(F::FreeMod, W::Vector{GrpAbFinGenElem})Given a free module F over a graded ring with grading group G, say, and given a vector W of ngens(F) elements of G, create a G-graded free module by assigning the entries of W as weights to the generators of F. Return the new module.
grade(F::FreeMod)As above, with all weights set to zero(G).
The function applies to free modules over both graded multivariate polynomial rings and their quotients.
Examples
julia> R, x, y = polynomial_ring(QQ, "x" => 1:2, "y" => 1:3);
julia> G = abelian_group([0, 0])
GrpAb: Z^2
julia> g = gens(G)
2-element Vector{GrpAbFinGenElem}:
Element of G with components [1 0]
Element of G with components [0 1]
julia> W = [g[1], g[1], g[2], g[2], g[2]];
julia> S, _ = grade(R, W)
(Graded multivariate polynomial ring in 5 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x[1], x[2], y[1], y[2], y[3]])
julia> F = free_module(S, 3)
Free module of rank 3 over S
julia> FF = grade(F)
Graded free module S^3([0 0]) of rank 3 over S
julia> F
Free module of rank 3 over Sgrade — Methodgrade(F::FreeMod, W::Vector{<:Vector{<:IntegerUnion}})Given a free module F over a graded ring with grading group $G = \mathbb Z^m$, and given a vector W of ngens(F) integer vectors of the same size m, say, define a $G$-grading on F by converting the vectors in W to elements of $G$, and assigning these elements as weights to the variables. Return the new module.
grade(F::FreeMod, W::Union{ZZMatrix, Matrix{<:IntegerUnion}})As above, converting the columns of W.
grade(F::FreeMod, W::Vector{<:IntegerUnion})Given a free module F over a graded ring with grading group $G = \mathbb Z$, and given a vector W of ngens(F) integers, define a $G$-grading on F converting the entries of W to elements of G, and assigning these elements as weights to the variables. Return the new module.
The function applies to free modules over both graded multivariate polynomial rings and their quotients.
Examples
julia> R, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"], [1 0 1; 0 1 1])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])
julia> F = free_module(R, 2)
Free module of rank 2 over R
julia> FF = grade(F, [[1, 0], [0, 1]])
Graded free module R^1([-1 0]) + R^1([0 -1]) of rank 2 over R
julia> FFF = grade(F, [1 0; 0 1])
Graded free module R^1([-1 0]) + R^1([0 -1]) of rank 2 over Rjulia> R, (x, y) = graded_polynomial_ring(QQ, ["x", "y"])
(Graded multivariate polynomial ring in 2 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y])
julia> S, _ = quo(R, [x*y])
(Quotient of multivariate polynomial ring by ideal with 1 generator, Map from
R to S defined by a julia-function with inverse)
julia> F = free_module(S, 2)
Free module of rank 2 over S
julia> FF = grade(F, [1, 2])
Graded free module S^1([-1]) + S^1([-2]) of rank 2 over Sgraded_free_module — Functiongraded_free_module(R::Ring, p::Int, W::Vector{GrpAbFinGenElem}=[grading_group(R)[0] for i in 1:p], name::String="e")Given a graded ring R with grading group G, say, and given a vector W with p elements of G, create the free module $R^p$ equipped with its basis of standard unit vectors, and assign weights to these vectors according to the entries of W. Return the resulting graded free module.
graded_free_module(R::Ring, W::Vector{GrpAbFinGenElem}, name::String="e")As above, with p = length(W).
The function applies to graded multivariate polynomial rings and their quotients.
The string name specifies how the basis vectors are printed.
Examples
julia> R, (x,y) = graded_polynomial_ring(QQ, ["x", "y"])
(Graded multivariate polynomial ring in 2 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y])
julia> graded_free_module(R,3)
Graded free module R^3([0]) of rank 3 over R
julia> G = grading_group(R)
GrpAb: Z
julia> graded_free_module(R, [G[1], 2*G[1]])
Graded free module R^1([-1]) + R^1([-2]) of rank 2 over Rgraded_free_module — Functiongraded_free_module(R::Ring, W::Vector{<:Vector{<:IntegerUnion}}, name::String="e")Given a graded ring R with grading group $G = \mathbb Z^m$, and given a vector W of integer vectors of the same size p, say, create the free module $R^p$ equipped with its basis of standard unit vectors, and assign weights to these vectors according to the entries of W, converted to elements of G. Return the resulting graded free module.
graded_free_module(R::Ring, W::Union{ZZMatrix, Matrix{<:IntegerUnion}}, name::String="e")As above, converting the columns of W.
graded_free_module(R::Ring, W::Vector{<:IntegerUnion}, name::String="e")Given a graded ring R with grading group $G = \mathbb Z$, and given a vector W of integers, set p = length(W), create the free module $R^p$ equipped with its basis of standard unit vectors, and assign weights to these vectors according to the entries of W, converted to elements of G. Return the resulting graded free module.
The string name specifies how the basis vectors are printed.
The function applies to graded multivariate polynomial rings and their quotients.
Examples
julia> R, (x,y) = graded_polynomial_ring(QQ, ["x", "y"]);
julia> F = graded_free_module(R, [1, 2])
Graded free module R^1([-1]) + R^1([-2]) of rank 2 over Rjulia> S, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"], [1 0 1; 0 1 1]);
julia> FF = graded_free_module(S, [[1, 2], [-1, 3]])
Graded free module S^1([-1 -2]) + S^1([1 -3]) of rank 2 over S
julia> FFF = graded_free_module(S, [1 -1; 2 3])
Graded free module S^1([-1 -2]) + S^1([1 -3]) of rank 2 over S
julia> FF == FFF
trueData Associated to Free Modules
If F is a free R-module, then
base_ring(F)refers toR,basis(F),gens(F)to the basis vectors ofF,rank(F),ngens(F),dim(F)to the number of these vectors, andF[i],basis(F, i),gen(F, i)to thei-th such vector.
Examples
julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);
julia> F = free_module(R, 3);
julia> basis(F)
3-element Vector{FreeModElem{QQMPolyRingElem}}:
e[1]
e[2]
e[3]
julia> rank(F)
3In the graded case, we also have:
grading_group — Methodgrading_group(F::FreeMod)Return the grading group of base_ring(F).
Examples
julia> R, (x,y) = graded_polynomial_ring(QQ, ["x", "y"]);
julia> F = graded_free_module(R, 3)
Graded free module R^3([0]) of rank 3 over R
julia> grading_group(F)
GrpAb: Zdegrees_of_generators — Methoddegrees_of_generators(F::FreeMod)Return the degrees of the generators of F.
Examples
julia> R, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"]);
julia> F = graded_free_module(R, 2)
Graded free module R^2([0]) of rank 2 over R
julia> degrees_of_generators(F)
2-element Vector{GrpAbFinGenElem}:
[0]
[0]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 TAlternatively, directly write the element as a linear combination of basis vectors of $F$:
Examples
julia> R, (x, y) = polynomial_ring(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
trueGiven an element f of a free module F over a multivariate polynomial ring with element type T,
parent(f)refers toF, andcoordinates(f)to the coordinate vector off, returned as an object of typeSRow{T}.
Examples
julia> R, (x, y) = polynomial_ring(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 2 variables over QQ
julia> coordinates(f)
Sparse row with positions [1, 3] and values QQMPolyRingElem[x, y]
The zero element of a free module is obtained as follows:
zero — Methodzero(F::AbstractFreeMod)Return the zero element of F.
Whether a given element of a free module is zero can be tested as follows:
is_zero — Methodis_zero(f::AbstractFreeModElem)Return true if f is zero, false otherwise.
In the graded case, we additionally have:
is_homogeneous — Methodis_homogeneous(f::FreeModElem)Given an element f of a graded free module, return true if f is homogeneous, false otherwise.
Examples
julia> R, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"], [1, 2, 3]);
julia> F = free_module(R, 2)
Free module of rank 2 over R
julia> FF = grade(F, [1,4])
Graded free module R^1([-1]) + R^1([-4]) of rank 2 over R
julia> f = y^2*2*FF[1]-x*FF[2]
2*y^2*e[1] - x*e[2]
julia> is_homogeneous(f)
truedegree — Methoddegree(f::FreeModElem)Given a homogeneous element f of a graded free module, return the degree of f.
degree(::Type{Vector{Int}}, f::FreeModElem)Given a homogeneous element f of a $\mathbb Z^m$-graded free module, return the degree of f, converted to a vector of integer numbers.
degree(::Type{Int}, f::FreeModElem)Given a homogeneous element f of a $\mathbb Z$-graded free module, return the degree of f, converted to an integer number.
Examples
julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
julia> f = y^2*z − x^2*w
-w*x^2 + y^2*z
julia> degree(f)
[3]
julia> typeof(degree(f))
GrpAbFinGenElem
julia> degree(Int, f)
3
julia> typeof(degree(Int, f))
Int64Tests on Free Modules
The tests is_graded, is_standard_graded, is_z_graded, and is_zm_graded carry over analogously to free modules. They return true if the corresponding property is satisfied, and false otherwise. In addition, we have:
== — Method==(F::FreeMod, G::FreeMod)Return true if F and G are equal, false otherwise.
Here, F and G are equal iff either
- both modules are ungraded and their base rings, ranks, and names for printing the basis elements are equal,
or else
- both modules are graded, the above holds, and for each $i$, the degrees of the $i$-th basis elements are equal.
is_isomorphic — Methodis_isomorphic(F::FreeMod, G::FreeMod)Return true if F and G are isomorphic as (graded) modules, false otherwise.
That is, either
- both modules are ungraded and their base rings and ranks are equal,
or else
- both modules are graded, the above holds, and the multisets of the degrees of the basis elements are equal.
Examples
julia> R, _= polynomial_ring(QQ, ["x", "y", "z"]);
julia> Z = abelian_group(0);
julia> Rg, (x, y, z)=grade(R,[Z[1], Z[1], Z[1]]);
julia> F = graded_free_module(Rg, [1,1,3,2]);
julia> G1 = graded_free_module(Rg, [1,1,2,3]);
julia> is_isomorphic(F, G1)
true
julia> G2 = graded_free_module(Rg, [1,1,5,6]);
julia> is_isomorphic(F, G2)
falseis_zero — Methodis_zero(F::AbstractFreeMod)Return true if F is the zero module, false otherwise.
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 $F\to M$ 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 — Methodhom(F::FreeMod, M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}) where TGiven 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 TGiven 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.
The module M may be of type FreeMod or SubquoMod. If both modules F and M are graded, the data must define a graded module homomorphism of some degree. If this degree is the zero element of the (common) grading group, we refer to the homomorphism under consideration as a homogeneous module homomorphism.
Examples
julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
julia> F = free_module(R, 3)
Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
julia> G = free_module(R, 2)
Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
3-element Vector{FreeModElem{QQMPolyRingElem}}:
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 3 variables over QQ
Codomain:
=========
Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
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 3 variables over QQ
Codomain:
=========
Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
julia> a == b
truejulia> R, _= polynomial_ring(QQ, ["x", "y", "z"]);
julia> Z = abelian_group(0);
julia> Rg, (x, y, z) = grade(R,[Z[1], Z[1], Z[1]]);
julia> F1 = graded_free_module(Rg, 3)
Graded free module Rg^3([0]) of rank 3 over Rg
julia> G1 = graded_free_module(Rg, 2)
Graded free module Rg^2([0]) of rank 2 over Rg
julia> V1 = [y*G1[1], (x+y)*G1[1]+y*G1[2], z*G1[2]]
3-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
y*e[1]
(x + y)*e[1] + y*e[2]
z*e[2]
julia> a1 = hom(F1, G1, V1)
F1 -> G1
e[1] -> y*e[1]
e[2] -> (x + y)*e[1] + y*e[2]
e[3] -> z*e[2]
Graded module homomorphism of degree [1]
julia> F2 = graded_free_module(Rg, [1,1,1])
Graded free module Rg^3([-1]) of rank 3 over Rg
julia> G2 = graded_free_module(Rg, [0,0])
Graded free module Rg^2([0]) of rank 2 over Rg
julia> V2 = [y*G2[1], (x+y)*G2[1]+y*G2[2], z*G2[2]]
3-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
y*e[1]
(x + y)*e[1] + y*e[2]
z*e[2]
julia> a2 = hom(F2, G2, V2)
F2 -> G2
e[1] -> y*e[1]
e[2] -> (x + y)*e[1] + y*e[2]
e[3] -> z*e[2]
Homogeneous module homomorphism
julia> B = Rg[y 0; x+y y; 0 z]
[ y 0]
[x + y y]
[ 0 z]
julia> b = hom(F2, G2, B)
F2 -> G2
e[1] -> y*e[1]
e[2] -> (x + y)*e[1] + y*e[2]
e[3] -> z*e[2]
Homogeneous module homomorphism
julia> a2 == b
truehom — Methodhom(F::FreeMod, M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}, h::RingMapType) where {T, RingMapType}Given a vector V of rank(F) elements of M and a ring map h from base_ring(F) to base_ring(M), return the base_ring(F)-homomorphism F $\to$ M which sends the i-th basis vector of F to the i-th entry of V, and the scalars in base_ring(F) to their images under h.
hom(F::FreeMod, M::ModuleFP{T}, A::MatElem{T}, h::RingMapType) where {T, RingMapType}Given a matrix A over base_ring(M) with rank(F) rows and ngens(M) columns and a ring map h from base_ring(F) to base_ring(M), return the base_ring(F)-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, and the scalars in base_ring(F) to their images under h.
The module M may be of type FreeMod or SubquoMod. If both modules F and M are graded, the data must define a graded module homomorphism of some degree. If this degree is the zero element of the (common) grading group, we refer to the homomorphism under consideration as a homogeneous module homomorphism.
Given a homomorphism of type FreeModuleHom, a matrix representing it is recovered by the following function:
matrix — Methodmatrix(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) = polynomial_ring(QQ, ["x", "y", "z"])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
julia> F = free_module(R, 3)
Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
julia> G = free_module(R, 2)
Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
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]The domain and codomain of a homomorphism a of type FreeModuleHom can be recovered by entering domain(a) and codomain(a), respectively.
The functions below test whether a homomorphism of type FreeModuleHom is graded and homogeneous, respectively.
is_graded — Methodis_graded(a::FreeModuleHom)Return true if a is graded, false otherwise.
Examples
julia> R, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"]);
julia> F = graded_free_module(R, 3)
Graded free module R^3([0]) of rank 3 over R
julia> G = graded_free_module(R, 2)
Graded free module R^2([0]) of rank 2 over R
julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
3-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
y*e[1]
x*e[1] + y*e[2]
z*e[2]
julia> a = hom(F, G, V)
F -> G
e[1] -> y*e[1]
e[2] -> x*e[1] + y*e[2]
e[3] -> z*e[2]
Graded module homomorphism of degree [1]
julia> is_graded(a)
trueis_homogeneous — Methodis_homogeneous(a::FreeModuleHom)Return true if a is homogeneous, false otherwise
Here, if G is the grading group of a, a is homogeneous if a is graded of degree zero(G).
Examples
julia> R, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"]);
julia> F = graded_free_module(R, 3)
Graded free module R^3([0]) of rank 3 over R
julia> G = graded_free_module(R, 2)
Graded free module R^2([0]) of rank 2 over R
julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
3-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
y*e[1]
x*e[1] + y*e[2]
z*e[2]
julia> a = hom(F, G, V)
F -> G
e[1] -> y*e[1]
e[2] -> x*e[1] + y*e[2]
e[3] -> z*e[2]
Graded module homomorphism of degree [1]
julia> is_homogeneous(a)
falseIn the graded case, we additionally have:
degree — Methoddegree(a::FreeModuleHom)If a is graded, return the degree of a.
Examples
julia> R, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"]);
julia> F = graded_free_module(R, 3)
Graded free module R^3([0]) of rank 3 over R
julia> G = graded_free_module(R, 2)
Graded free module R^2([0]) of rank 2 over R
julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
3-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
y*e[1]
x*e[1] + y*e[2]
z*e[2]
julia> a = hom(F, G, V)
F -> G
e[1] -> y*e[1]
e[2] -> x*e[1] + y*e[2]
e[3] -> z*e[2]
Graded module homomorphism of degree [1]
julia> degree(a)
[1]grading_group — Methodgrading_group(a::FreeModuleHom)If a is graded, return the grading group of a.
Examples
julia> R, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"]);
julia> F = graded_free_module(R, 3)
Graded free module R^3([0]) of rank 3 over R
julia> G = graded_free_module(R, 2)
Graded free module R^2([0]) of rank 2 over R
julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
3-element Vector{FreeModElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
y*e[1]
x*e[1] + y*e[2]
z*e[2]
julia> a = hom(F, G, V)
F -> G
e[1] -> y*e[1]
e[2] -> x*e[1] + y*e[2]
e[3] -> z*e[2]
Graded module homomorphism of degree [1]
julia> is_graded(a)
true
julia> grading_group(a)
GrpAb: Z