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mpolyquo-localizations.jl
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mpolyquo-localizations.jl
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import AbstractAlgebra: Ring, RingElem, Generic.Frac
import Base: issubset
########################################################################
# Localizations of polynomial algebras #
########################################################################
#
# Let R = 𝕜[x₁,…,xₘ] be a polynomial ring, I ⊂ R some ideal
# and P = R/I its quotient. Then P is naturally an R-module
# and localization of P as a ring coincides with localization
# as an R-module in the sense that for every multiplicative
# set T ⊂ R there is a commutative diagram
#
# R → P = R/I
# ↓ ↓
# W = R[T⁻¹] → P[T⁻¹].
#
# Observe that, moreover, for every multiplicative set
# T' ⊂ P the preimage T of T' in R is also a multiplicative set.
#
# We may therefore treat localizations of polynomial algebras
# as localizations of modules over free polynomial rings and
# exclusively use the types of multiplicative sets which are
# available for the latter.
#
# Note the following differences compared to the standard usage
# of the localization interface:
#
# * The `base_ring` returns neither P, nor W, but R.
# * The `BaseRingType` is the type of R and similar for
# the other ring-based type parameters.
#
# This is to make the data structure most accessible for
# the computational backends.
#
# * The type returned by `numerator` and `denominator`
# on an element of type `MPolyQuoLocRingElem` is
# not `RingElemType`, but the type of `P`.
#
# This is to comply with the purely mathematical viewpoint
# where elements of localized rings are fractions of
# residue classes rather than residue classes of fractions.
#
@doc raw"""
MPolyQuoLocRing{
BaseRingType,
BaseRingElemType,
RingType,
RingElemType,
MultSetType <: AbsMultSet{RingType, RingElemType}
} <: AbsLocalizedRing{
RingType,
RingElemType,
MultSetType
}
Localization ``L = (𝕜[x₁,…,xₙ]/I)[S⁻¹]`` of a quotient
``𝕜[x₁,…,xₙ]/I`` of a polynomial ring ``P = 𝕜[x₁,…,xₙ]``
of type `RingType` over a base ring ``𝕜`` of type `BaseRingType` at a
multiplicative set ``S ⊂ P`` of type `MultSetType`.
"""
@attributes mutable struct MPolyQuoLocRing{
BaseRingType,
BaseRingElemType,
RingType,
RingElemType,
MultSetType <: AbsMultSet{RingType, RingElemType}
} <: AbsLocalizedRing{
RingType,
RingElemType,
MultSetType
}
R::RingType
I::MPolyIdeal{RingElemType}
S::MultSetType
Q::MPolyQuoRing{RingElemType}
W::MPolyLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
function MPolyQuoLocRing(
R::RingType,
I::MPolyIdeal{RingElemType},
S::MultSetType,
Q::MPolyQuoRing{RingElemType},
W::MPolyLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
) where {
BaseRingType<:Ring,
BaseRingElemType<:RingElem,
RingType<:MPolyRing,
RingElemType<:MPolyRingElem,
MultSetType<:AbsMultSet{RingType, RingElemType}
}
base_ring(I) == R || error("Ideal does not belong to the ring")
base_ring(Q) == R || error("The quotient ring does not come from the given ring")
# The following line throws obscure error messages that might yield a bug for MPolyIdeals.
# So it's commented out for now.
#modulus(Q) == I || error("the modulus of the quotient ring does not coincide with the ideal")
S == inverted_set(W) || error("the multiplicative set does not coincide with the inverted set of the localized ring")
base_ring(W) == R || error("the localization does not come from the given ring")
ambient_ring(S) == R || error("Multiplicative set does not belong to the ring")
k = coefficient_ring(R)
L = new{typeof(k), elem_type(k), typeof(R), RingElemType, MultSetType}(R, I, S, Q, W)
return L
end
end
### for convenience of later use
MPAnyQuoRing = Union{MPolyQuoLocRing,
MPolyQuoRing
}
MPAnyNonQuoRing = Union{MPolyRing, MPolyLocRing
}
MPolyAnyRing = Union{MPolyRing, MPolyQuoRing,
MPolyLocRing,MPolyQuoLocRing
}
### type getters
coefficient_ring_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = BRT
coefficient_ring_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = coefficient_ring_type(typeof(L))
coefficient_ring_elem_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = BRET
coefficient_ring_elem_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = coefficient_ring_elem_type(typeof(L))
base_ring_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = RT
base_ring_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_type(typeof(L))
base_ring_elem_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = RET
base_ring_elem_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_elem_type(typeof(L))
mult_set_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = MST
mult_set_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = mult_set_type(typeof(L))
localized_ring_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = MPolyLocRing{BRT, BRET, RT, RET, MST}
localized_ring_type(L::MPolyQuoLocRing) = localized_ring_type(typeof(L))
ideal_type(::Type{MPolyQuoLocalizedRingType}) where {MPolyQuoLocalizedRingType<:MPolyQuoLocRing} = MPolyQuoLocalizedIdeal{MPolyQuoLocalizedRingType, elem_type(MPolyQuoLocalizedRingType), ideal_type(localized_ring_type(MPolyQuoLocalizedRingType))}
ideal_type(W::MPolyQuoLocRing) = ideal_type(typeof(W))
### required getter functions
base_ring(L::MPolyQuoLocRing) = L.R
inverted_set(L::MPolyQuoLocRing) = L.S
### additional getter functions
@doc raw"""
modulus(L::MPolyQuoLocRing)
Given ``L = (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return ``IS⁻¹``.
"""
function modulus(L::MPolyQuoLocRing)
if !has_attribute(L, :modulus)
set_attribute!(L, :modulus, localized_ring(L)(L.I))
end
return get_attribute(L, :modulus)::ideal_type(localized_ring_type(L))
end
### for compatibility -- also provide modulus in the trivial case
modulus(R::MPAnyNonQuoRing)=ideal(R, elem_type(R)[])
@doc raw"""
underlying_quotient(L::MPolyQuoLocRing)
Given ``L = (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return ``𝕜[x₁,…,xₙ]/I``.
"""
underlying_quotient(L::MPolyQuoLocRing) = L.Q
## 3 more signatures for compatibility to make quotient_ring agnostic
underlying_quotient(L::MPolyQuoRing) = L
@attr MPolyQuoRing function underlying_quotient(L::MPolyRing)
return quo(L,ideal(L,[zero(L)]))[1]
end
@attr MPolyQuoRing function underlying_quotient(L::MPolyLocRing)
P = base_ring(L)
return quo(P,ideal(P,[zero(P)]))[1]
end
@doc raw"""
localized_ring(L::MPolyQuoLocRing)
Given ``L = (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return ``𝕜[x₁,…,xₙ][S⁻¹]``.
"""
localized_ring(L::MPolyQuoLocRing) = L.W
## 3 more signatures for compatibility to make localized_ring agnostic
localized_ring(L::MPolyLocRing) = L
@attr MPolyLocRing function localized_ring(L::MPolyQuoRing)
P = base_ring(L)
return localization(P, units_of(P))[1]
end
@attr MPolyLocRing function localized_ring(L::MPolyRing)
return localization(L, units_of(L))[1]
end
@doc raw"""
gens(L::MPolyQuoLocRing)
Given ``L = (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return the vector ``[x₁//1,…,xₙ//1]∈ Lⁿ``.
"""
gens(L::MPolyQuoLocRing) = L.(gens(base_ring(L)))
gen(L::MPolyQuoLocRing, i::Int) = L(gen(base_ring(L), i))
### printing
function Base.show(io::IO, ::MIME"text/plain", L::MPolyQuoLocRing)
io = pretty(io)
println(io, "Localization")
print(io, Indent())
println(io, "of ", Lowercase(), underlying_quotient(L))
print(io, "at ", Lowercase(), inverted_set(L))
print(io, Dedent())
end
function Base.show(io::IO, L::MPolyQuoLocRing)
io = pretty(io)
if get(io, :supercompact, false)
print(io, "Localized quotient of multivariate polynomial ring")
else
io = pretty(IOContext(io, :supercompact=>true))
print(io, "Localization of ")
print(io, Lowercase(), underlying_quotient(L))
print(io, " at ", Lowercase(), inverted_set(L))
end
end
### additional constructors
function quo(
W::MPolyLocRing,
I::MPolyLocalizedIdeal
)
R = base_ring(W)
S = inverted_set(W)
J = ideal(R, numerator.(gens(I)))
L = MPolyQuoLocRing(R, J, S, quo(R, J)[1], W)
return L, hom(W, L, hom(R, L, gens(L), check=false), check=false)
end
function quo(
L::MPolyQuoLocRing,
I::MPolyLocalizedIdeal
)
R = base_ring(L)
S = inverted_set(L)
base_ring(I) = localized_ring(L) || error("ideal does not belong to the correct ring")
J = pre_saturated_ideal(I)
W = MPolyQuoLocRing(R, J, S, quo(R, J)[1], localized_ring(L))
return W, hom(L, W, gens(W), check=false)
end
function quo(
L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST},
J::MPolyIdeal{RET}
) where {BRT, BRET, RT, RET, MST}
R = base_ring(L)
S = inverted_set(L)
W = localized_ring(L)
J = J + modulus(underlying_quotient(L))
P = MPolyQuoLocRing(R, J, S, quo(R, J)[1], W)
return P, hom(L, P, gens(P), check=false)
end
@doc raw"""
localization(RQ::MPolyQuoRing, U::AbsMPolyMultSet)
Given a quotient `RQ` of a multivariate polynomial ring `R` with projection map
`p : R -> RQ`, say, and given a multiplicatively closed subset `U` of `R`, return the
localization of `RQ` at `p(U)`, together with the localization map.
# Examples
```jldoctest
julia> T, t = polynomial_ring(QQ, "t");
julia> K, a = number_field(2*t^2-1, "a");
julia> R, (x, y) = polynomial_ring(K, ["x", "y"]);
julia> I = ideal(R, [2*x^2-y^3, 2*x^2-y^5])
ideal(2*x^2 - y^3, 2*x^2 - y^5)
julia> P = ideal(R, [y-1, x-a])
ideal(y - 1, x - a)
julia> U = complement_of_prime_ideal(P)
Complement
of prime ideal(y - 1, x - a)
in multivariate polynomial ring in 2 variables over number field
julia> RQ, _ = quo(R, I);
julia> RQL, iota = localization(RQ, U);
julia> RQL
Localization
of quotient of multivariate polynomial ring by ideal with 2 generators
at complement of prime ideal(y - 1, x - a)
julia> iota
Map from
RQ to Localization of quotient of multivariate polynomial ring at complement of prime ideal defined by a julia-function
```
""" localization(A::MPolyQuoRing, U::AbsMPolyMultSet)
###localization is an Abstract Algebra alias for Localization
function Localization(Q::MPolyQuoRing{RET}, S::MultSetType) where {RET <: RingElem, MultSetType <: AbsMultSet}
L = MPolyQuoLocRing(base_ring(Q), modulus(Q), S, Q, Localization(S)[1])
return L, MapFromFunc((x->L(lift(x))), Q, L)
end
function Localization(
L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST},
S::AbsMPolyMultSet{BRT, BRET, RT, RET}
) where {BRT, BRET, RT, RET, MST}
ambient_ring(S) == base_ring(L) || error("multiplicative set does not belong to the correct ring")
issubset(S, inverted_set(L)) && return L, MapFromFunc(x->x, L, L)
U = inverted_set(L)*S
W = MPolyQuoLocRing(base_ring(L), modulus(underlying_quotient(L)), U, underlying_quotient(L), Localization(U)[1])
return W, MapFromFunc((x->W(lifted_numerator(x), lifted_denominator(x), check=false)), L, W)
end
function MPolyQuoLocRing(R::RT, I::Ideal{RET}, T::MultSetType) where {RT<:MPolyRing, RET<:MPolyRingElem, MultSetType<:AbsMultSet}
return MPolyQuoLocRing(R, I, T, quo(R, I)[1], Localization(T)[1])
end
function MPolyQuoLocRing(R::RT) where {RT<:MPolyRing}
I = ideal(R, zero(R))
Q, _ = quo(R, I)
U = units_of(R)
W, _ = Localization(U)
return MPolyQuoLocRing(R, I, U, Q, W)
end
function MPolyQuoLocRing(Q::RT) where {RT<:MPolyQuoRing}
R = base_ring(Q)
I = modulus(Q)
U = units_of(R)
W, _ = Localization(U)
return MPolyQuoLocRing(R, I, U, Q, W)
end
function MPolyQuoLocRing(W::MPolyLocRing)
R = base_ring(W)
I = ideal(R, zero(R))
Q, _ = quo(R, I)
U = inverted_set(W)
return MPolyQuoLocRing(R, I, U, Q, W)
end
function Base.in(f::AbstractAlgebra.Generic.Frac{RET}, L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST}
R = base_ring(L)
R == parent(numerator(f)) || error("element does not belong to the correct ring")
denominator(f) in inverted_set(L) && return true
return numerator(f) in ideal(L, denominator(f))
end
### generation of random elements
function rand(W::MPolyQuoLocRing, v1::UnitRange{Int}, v2::UnitRange{Int}, v3::UnitRange{Int})
return W(rand(localized_ring(W), v1, v2, v3))
end
########################################################################
# Elements of localizations of polynomial algebras #
########################################################################
@doc raw"""
MPolyQuoLocRingElem{
BaseRingType,
BaseRingElemType,
RingType,
RingElemType,
MultSetType
} <: AbsLocalizedRingElem{
RingType,
RingElemType,
MultSetType
}
Elements ``a//b`` of localizations ``L = (𝕜[x₁,…,xₙ]/I)[S⁻¹]`` of type
`MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}`.
"""
mutable struct MPolyQuoLocRingElem{
BaseRingType,
BaseRingElemType,
RingType,
RingElemType,
MultSetType
} <: AbsLocalizedRingElem{
RingType,
RingElemType,
MultSetType
}
# the parent ring
L::MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
# representatives of numerator and denominator
numerator::RingElemType
denominator::RingElemType
is_reduced::Bool
function MPolyQuoLocRingElem(
L::MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType},
a::RingElemType,
b::RingElemType;
check::Bool=true,
is_reduced::Bool=false
) where {BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
S = inverted_set(L)
R = base_ring(L)
parent(a) == parent(b) == R || error("elements do not belong to the correct ring")
check && (b in S || error("denominator is not admissible"))
return new{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}(L, a, b, is_reduced)
end
end
### type getters
coefficient_ring_type(::Type{MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = BRT
coefficient_ring_type(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_type(typeof(f))
coefficient_ring_elem_type(::Type{MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = BRET
coefficient_ring_elem_type(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_type(typeof(f))
base_ring_type(::Type{MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = RT
base_ring_type(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_type(typeof(f))
base_ring_elem_type(::Type{MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = RET
base_ring_elem_type(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_type(typeof(f))
mult_set_type(::Type{MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = MST
mult_set_type(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_type(typeof(f))
### required getter functions
parent(a::MPolyQuoLocRingElem) = a.L
numerator(a::MPolyQuoLocRingElem) = underlying_quotient(parent(a))(a.numerator)
denominator(a::MPolyQuoLocRingElem) = underlying_quotient(parent(a))(a.denominator)
### additional getter functions
underlying_quotient(a::MPolyQuoLocRingElem) = underlying_quotient(parent(a))
localized_ring(a::MPolyQuoLocRingElem) = localized_ring(parent(a))
base_ring(a::MPolyQuoLocRingElem) = base_ring(parent(a))
is_reduced(a::MPolyQuoLocRingElem) = a.is_reduced
@doc raw"""
lifted_numerator(a::MPolyQuoLocRingElem)
Given ``A//B ∈ (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return a representative
``a ∈ 𝕜[x₁,…,xₙ]`` of the numerator.
"""
lifted_numerator(a::MPolyQuoLocRingElem) = a.numerator
@doc raw"""
lifted_denominator(a::MPolyQuoLocRingElem)
Given ``A//B ∈ (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return a representative
``b ∈ 𝕜[x₁,…,xₙ]`` of the denominator.
"""
lifted_denominator(a::MPolyQuoLocRingElem) = a.denominator
@doc raw"""
fraction(a::MPolyQuoLocRingElem)
Given ``A//B ∈ (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return a representative
``a//b ∈ Quot(𝕜[x₁,…,xₙ])`` of the fraction.
"""
fraction(a::MPolyQuoLocRingElem) = lifted_numerator(a)//lifted_denominator(a)
### copying of elements
function Base.deepcopy_internal(f::MPolyQuoLocRingElem, dict::IdDict)
return parent(f)(f, check=false, is_reduced=is_reduced(f))
end
### required conversions
(L::MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType})(f::RingElemType, is_reduced::Bool=false) where {BaseRingType, BaseRingElemType, RingType, RingElemType<:RingElem, MultSetType} = MPolyQuoLocRingElem(L, f, one(f), check=false, is_reduced=is_reduced)
function (L::MPolyQuoLocRing{
BaseRingType,
BaseRingElemType,
RingType,
RingElemType,
MultSetType
})(
a::RingElemType,
b::RingElemType;
check::Bool=true,
is_reduced::Bool=false
) where {
BaseRingType,
BaseRingElemType,
RingType,
RingElemType,
MultSetType
}
check || return MPolyQuoLocRingElem(L, a, b, check=false, is_reduced=is_reduced)
b in inverted_set(L) || return convert(L, a//b)
return MPolyQuoLocRingElem(L, a, b, check=false, is_reduced=is_reduced)
end
### additional conversions
function (L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST})(f::Frac{RET}; check::Bool=true, is_reduced::Bool=false) where {BRT, BRET, RT, RET, MST}
R = base_ring(L)
return L(R(numerator(f)), R(denominator(f)), check=check, is_reduced=is_reduced)
end
(L::MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType})(a::T, b::T; check::Bool=true, is_reduced::Bool=false) where {BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType, T<:MPolyQuoRingElem{RingElemType}} = L(lift(a), lift(b), check=check, is_reduced=is_reduced)
function (L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST})(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}; check::Bool=true, is_reduced::Bool=false) where {BRT, BRET, RT, RET, MST}
parent(f) === L && return f
return L(lifted_numerator(f), lifted_denominator(f), check=check, is_reduced=is_reduced)
end
function (L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST})(f::MPolyLocRingElem{BRT, BRET, RT, RET, MST}; check::Bool=true, is_reduced::Bool=false) where {BRT, BRET, RT, RET, MST}
parent(f) === localized_ring(L) && return L(numerator(f), denominator(f), check=false, is_reduced=is_reduced)
return L(numerator(f), denominator(f), check=check, is_reduced=is_reduced)
end
function (L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST})(f::MPolyLocRingElem{BRT, BRET, RT, RET, MST}; check::Bool=true, is_reduced::Bool=false) where {BRT, BRET, RT, RET, MST<:MPolyComplementOfKPointIdeal}
return L(numerator(f), denominator(f), check=check, is_reduced=is_reduced)
end
function (L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST})(f::MPolyQuoRingElem{RET}; check::Bool=true, is_reduced::Bool=false) where {BRT, BRET, RT, RET, MST}
base_ring(parent(f)) == base_ring(L) || error("the given element does not belong to the correct ring")
check && (parent(f) == underlying_quotient(L) || all(x->(iszero(L(x))), gens(modulus(parent(f)))) || error("coercion is not well defined"))
return L(lift(f))
end
### additional functionality
@doc raw"""
lift(f::MPolyQuoLocRingElem)
Given ``f = A//B ∈ (𝕜[x₁,…,xₙ]/I)[S⁻¹]``, return a representative
``a//b ∈ 𝕜[x₁,…,xₙ][S⁻¹]`` of the fraction.
"""
lift(f::MPolyQuoLocRingElem) = localized_ring(f)(lifted_numerator(f), lifted_denominator(f))
@doc raw"""
is_unit(f::MPolyQuoLocRingElem)
Return `true`, if `f` is a unit of `parent(f)`, `true` otherwise.
# Examples
```jldoctest
julia> T, t = polynomial_ring(QQ, "t");
julia> K, a = number_field(2*t^2-1, "a");
julia> R, (x, y) = polynomial_ring(K, ["x", "y"]);
julia> I = ideal(R, [2*x^2-y^3, 2*x^2-y^5])
ideal(2*x^2 - y^3, 2*x^2 - y^5)
julia> P = ideal(R, [y-1, x-a])
ideal(y - 1, x - a)
julia> U = complement_of_prime_ideal(P)
Complement
of prime ideal(y - 1, x - a)
in multivariate polynomial ring in 2 variables over number field
julia> RQ, p = quo(R, I);
julia> RQL, iota = Localization(RQ, U);
julia> is_unit(iota(p(x)))
true
```
""" is_unit(f::MPolyQuoLocRingElem)
function is_unit(f::MPolyQuoLocRingElem)
lifted_numerator(f) in inverted_set(parent(f)) && return true
R=localized_ring(parent(f))
return one(R) in modulus(parent(f)) + ideal(R, lift(f))
end
function is_unit(L::MPolyQuoLocRing, f::MPolyLocRingElem)
parent(f) == localized_ring(L) || error("element does not belong to the correct ring")
numerator(f) in inverted_set(L) && return true
one(localized_ring(L)) in modulus(L) + ideal(localized_ring(L), f)
end
function is_unit(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}, f::RET) where {BRT, BRET, RT, RET, MST}
parent(f) == base_ring(L) || error("element does not belong to the correct ring")
f in inverted_set(L) && return true
return one(localized_ring(L)) in modulus(L) + ideal(localized_ring(L), localized_ring(L)(f))
end
function is_unit(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}, f::MPolyQuoRingElem{RET}) where {BRT, BRET, RT, RET, MST}
parent(f) == underlying_quotient(L) || error("element does not belong to the correct ring")
lift(f) in inverted_set(L) && return true
one(localized_ring(L)) in modulus(L) + ideal(localized_ring(L), localized_ring(L)(f))
end
# WARNING: This routine runs forever if f is not a unit in L.
# So this needs to be checked first!
function inv(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MPolyPowersOfElement{BRT, BRET, RT, RET}},
f::MPolyQuoRingElem{RET}) where {BRT, BRET, RT, RET}
Q = underlying_quotient(L)
parent(f) == underlying_quotient(L) || error("element does not belong to the correct ring")
W = localized_ring(L)
R = base_ring(L)
I = saturated_ideal(modulus(L))
d = prod(denominators(inverted_set(W)))
powers_of_d = [d]
### apply logarithmic bisection to find a power dᵏ ≡ c ⋅ f mod I
(result, coefficient) = divides(one(Q), f)
# check whether f is already a unit
result && return L(coefficient)
push!(powers_of_d, d)
abort = false
# find some power which works
while !abort
(abort, coefficient) = divides(Q(last(powers_of_d)), f)
if !abort
push!(powers_of_d, last(powers_of_d)^2)
end
end
# find the minimal power that works
upper = pop!(powers_of_d)
lower = pop!(powers_of_d)
while length(powers_of_d) > 0
middle = lower*pop!(powers_of_d)
(result, coefficient) = divides(Q(middle), f)
if result
upper = middle
else
lower = middle
end
end
return L(lift(coefficient), upper)
end
function inv(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MPolyPowersOfElement{BRT, BRET, RT, RET}}) where {BRT, BRET, RT, RET}
return parent(f)(denominator(f), numerator(f))
end
###
# Assume that [f] = [a]//[b] is an admissible element of L = (R/I)[S⁻¹] and bring it
# to the form [f] = [c]//[dᵏ] with d∈ S.
function convert(
L::MPolyQuoLocRing{BRT, BRET, RT, RET, MPolyPowersOfElement{BRT, BRET, RT, RET}},
f::AbstractAlgebra.Generic.Frac{RET}
) where {BRT, BRET, RT, RET}
a = numerator(f)
b = denominator(f)
Q = underlying_quotient(L)
parent(a) == base_ring(L) || error("element does not belong to the correct ring")
W = localized_ring(L)
R = base_ring(L)
I = saturated_ideal(modulus(L))
one(R) in I && return zero(L)
d = prod(denominators(inverted_set(W)))
powers_of_d = [d]
### apply logarithmic bisection to find a power a ⋅dᵏ ≡ c ⋅ b mod I
(result, coefficient) = divides(Q(a), Q(b))
# check whether f is already a unit
result && return L(coefficient)
push!(powers_of_d, d)
abort = false
# find some power which works
while !abort
(abort, coefficient) = divides(Q(a*last(powers_of_d)), Q(b))
if !abort
push!(powers_of_d, last(powers_of_d)^2)
end
end
# find the minimal power that works
upper = pop!(powers_of_d)
lower = pop!(powers_of_d)
while length(powers_of_d) > 0
middle = lower*pop!(powers_of_d)
(result, coefficient) = divides(Q(a*middle), Q(b))
if result
upper = middle
else
lower = middle
end
end
return L(lift(coefficient), upper)
end
function _is_regular_fraction(R::RingType, p::MPolyRingElem, q::MPolyRingElem) where {RingType<:Union{MPolyQuoLocRing, MPolyLocRing, MPolyRing, MPolyQuoRing}}
return divides(R(p), R(q))[1]
end
### Extensions for coherence
#function _is_regular_fraction(R::MPolyLocRing, p::MPolyRingElem, q::MPolyRingElem)
# return divides(R(p), R(q))[1]
#end
#
#function _is_regular_fraction(R::MPolyRing, p::MPolyRingElem, q::MPolyRingElem)
# return divides(p, q)[1]
#end
#
#function _is_regular_fraction(R::MPolyQuoRing, p::MPolyRingElem, q::MPolyRingElem)
# return divides(R(p), R(q))[1]
#end
### arithmetic #########################################################
function +(a::T, b::T) where {T<:MPolyQuoLocRingElem}
parent(a) == parent(b) || error("the arguments do not have the same parent ring")
if lifted_denominator(a) == lifted_denominator(b)
return (parent(a))(lifted_numerator(a) + lifted_numerator(b), lifted_denominator(a), check=false)
end
return (parent(a))(lifted_numerator(a)*lifted_denominator(b) + lifted_numerator(b)*lifted_denominator(a), lifted_denominator(a)*lifted_denominator(b), check=false)
end
# TODO: improve this method.
function addeq!(a::T, b::T) where {T<:MPolyQuoLocRingElem}
a = a+b
return a
end
function -(a::T, b::T) where {T<:MPolyQuoLocRingElem}
parent(a) == parent(b) || error("the arguments do not have the same parent ring")
if lifted_denominator(a) == lifted_denominator(b)
return (parent(a))(lifted_numerator(a) - lifted_numerator(b), lifted_denominator(a), check=false)
end
return (parent(a))(lifted_numerator(a)*lifted_denominator(b) - lifted_numerator(b)*lifted_denominator(a), lifted_denominator(a)*lifted_denominator(b), check=false)
end
function *(a::T, b::T) where {T<:MPolyQuoLocRingElem}
parent(a) == parent(b) || error("the arguments do not have the same parent ring")
return (parent(a))(lifted_numerator(a)*lifted_numerator(b), lifted_denominator(a)*lifted_denominator(b), check=false)
end
function *(a::RET, b::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT<:Ring, BRET<:RingElem, RT<:Ring, RET <: RingElem, MST}
return (parent(b))(a*lifted_numerator(b), lifted_denominator(b), check=false)
end
function *(a::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}, b::RET) where {BRT<:Ring, BRET<:RingElem, RT<:Ring, RET <: RingElem, MST}
return b*a
end
function *(a::BRET, b::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT<:Ring, BRET<:RingElem, RT<:Ring, RET <: RingElem, MST}
return (parent(b))(a*lifted_numerator(b), lifted_denominator(b), check=false)
end
function *(a::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}, b::BRET) where {BRT<:Ring, BRET<:RingElem, RT<:Ring, RET <: RingElem, MST}
return b*a
end
### Why are the `//`-methods not implemented?
# Since a quotient ring Q = R/I of a polynomial ring R is not necessarily
# factorial, it is difficult to decide, whether or not a and b have a
# common factor g that can be cancelled so that b'= b/g ∈ Q belongs
# to the multiplicative set. Moreover, this would be the case if any
# lift of b' belonged to S + I where S ⊂ R is the original multiplicative
# set. Such containment can not easily be checked based only on the
# functionality provided for S: Depending on the concrete type of
# S, this task is algorithmically difficult, if not impossible.
#
# To remedy for this, we pursue the following pattern:
#
# * Creation of elements [a]/[b] ∈ Q[S⁻¹] is possible only from
# representatives a/b ∈ R[S⁻¹] with b ∈ S.
# * The ring arithmetic attempts to cancel fractions which includes
# reduction modulo I of both the numerator and the denominator.
# This leads to representatives which would not be admissible
# for creation of elements in Q[S⁻¹].
# * Division routines can be used for the ring R[S⁻¹] with subsequent
# conversion.
function Base.:(/)(a::Oscar.IntegerUnion, b::MPolyQuoLocRingElem)
success, c = divides(parent(b), b)
!success && error("$b does not divide $a")
return c
end
function Base.:(/)(a::T, b::T) where {T<:MPolyQuoLocRingElem}
success, c = divides(a, b)
!success && error("$b does not divide $a")
return c
end
function divexact(a::Oscar.IntegerUnion, b::MPolyQuoLocRingElem)
return a/b
end
function divexact(a::T, b::T) where {T<:MPolyQuoLocRingElem}
return a/b
end
function ==(a::T, b::T) where {T<:MPolyQuoLocRingElem}
parent(a) == parent(b) || error("the arguments do not have the same parent ring")
return lifted_numerator(a)*lifted_denominator(b) - lifted_numerator(b)*lifted_denominator(a) in modulus(parent(a))
end
function ^(a::MPolyQuoLocRingElem, i::ZZRingElem)
return parent(a)(lifted_numerator(a)^i, lifted_denominator(a)^i, check=false)
end
function ^(a::MPolyQuoLocRingElem, i::Integer)
return parent(a)(lifted_numerator(a)^i, lifted_denominator(a)^i, check=false)
end
function isone(a::MPolyQuoLocRingElem)
return lifted_numerator(a) - lifted_denominator(a) in modulus(parent(a))
end
function iszero(a::MPolyQuoLocRingElem)
return lift(a) in modulus(parent(a))
end
### enhancement of the arithmetic
function reduce_fraction(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST<:MPolyPowersOfElement}
return f # Disable reduction here, because it slows down arithmetic.
return parent(f)(lift(simplify(numerator(f))), lifted_denominator(f), check=false)
end
# for local orderings, reduction does not give the correct result.
function reduce_fraction(f::MPolyQuoLocRingElem{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST<:MPolyComplementOfKPointIdeal}
is_reduced(f) && return f
return f
end
### implementation of Oscar's general ring interface
one(W::MPolyQuoLocRing) = W(one(base_ring(W)))
zero(W::MPolyQuoLocRing)= W(zero(base_ring(W)))
elem_type(W::MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}) where {BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType} = MPolyQuoLocRingElem{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
elem_type(T::Type{MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}}) where {BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType} = MPolyQuoLocRingElem{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
parent_type(W::MPolyQuoLocRingElem{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}) where {BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType} = MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
parent_type(T::Type{MPolyQuoLocRingElem{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}}) where {BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType} = MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, MultSetType}
@doc raw"""
bring_to_common_denominator(f::Vector{T}) where {T<:MPolyQuoLocRingElem}
Given a vector of fractions ``[a₁//b₁,…,aₙ//bₙ]`` return a pair
``(d, λ)`` consisting of a common denominator ``d`` and a vector
``λ = [λ₁,…,λₙ]`` such that ``aᵢ//bᵢ = λᵢ⋅aᵢ//d``.
"""
function bring_to_common_denominator(f::Vector{T}) where {T<:MPolyQuoLocRingElem}
length(f) == 0 && error("need at least one argument to determine the return type")
R = base_ring(parent(f[1]))
for a in f
R == base_ring(parent(a)) || error("elements do not belong to the same ring")
end
d = one(R)
a = Vector{elem_type(R)}()
for den in lifted_denominator.(f)
b = gcd(d, den)
c = divexact(den, b)
e = divexact(d, b)
d = d*c
a = [c*k for k in a]
push!(a, e)
end
return d, a
end
@doc raw"""
write_as_linear_combination(f::T, g::Vector{T}) where {T<:MPolyLocRingElem}
Write ``f = ∑ᵢ λᵢ⋅gᵢ`` for some ``λᵢ`` and return the vector ``[λ₁,…,λₙ]``.
"""
function write_as_linear_combination(
f::RingElemType,
g::Vector{RingElemType}
) where {RingElemType<:MPolyQuoLocRingElem}
n = length(g)
L = parent(f)
W = localized_ring(L)
for a in g
parent(a) == L || error("elements do not belong to the same ring")
end
return L.(vec(coordinates(lift(f), ideal(L, g)))[1:length(g)]) # temporary hack; to be replaced.
end
write_as_linear_combination(f::MPolyQuoLocRingElem, g::Vector) = write_as_linear_combination(f, parent(f).(g))
########################################################################
# Homomorphisms of quotients of localized polynomial algebras #
########################################################################
#
# Suppose we are given two localizations of polynomial algebras
# by means of commutative diagrams
#
# R → P = R/I
# ↓ ↓
# V = R[T⁻¹] → P[T⁻¹]
#
# and
#
# S → Q = S/J
# ↓ ↓
# W = S[U⁻¹] → Q[U⁻¹].
#
# Lemma:
# For any homomorphism φ : P[T⁻¹] → Q[U⁻¹] the following holds.
#
# φ
# P[T⁻¹] → Q[U⁻¹]
# ↑ ↑
# R[T⁻¹] --> S[U⁻¹]
# ↑ ↗ ψ ↑ ι
# R → S[c⁻¹]
# η ↑ κ
# S
#
# a) The composition of maps R → Q[U⁻¹] completely determines φ by
# the images xᵢ ↦ [aᵢ]/[bᵢ] with aᵢ ∈ S, bᵢ ∈ U.
# b) Let ψ : R → S[U⁻¹] be the map determined by some choice of
# the images xᵢ↦ aᵢ/bᵢ as above. Then ψ extends to a map
# R[T⁻¹] → S[U⁻¹] if and only if
#
# for all t ∈ T : ψ(t) ∈ U.
#
# This is not necessarily the case as the lift of images
# φ(t) ∈ Q[U⁻¹] in S[U⁻¹] need only be elements of U + J.
# c) Choosing a common denominator c for all ψ(xᵢ), we obtain a
# ring homomorphism η : R → S[c⁻¹] such that ψ = ι ∘ η.
#
# Upshot: In order to describe φ, we may store some homomorphism
#
# ψ : R → S[U⁻¹]
#
# lifting it and keep in mind the ambiguity of choices for such ψ.
# The latter point c) will be useful for reducing to a homomorphism
# of finitely generated algebras.
@doc raw"""
MPolyQuoLocalizedRingHom{
BaseRingType,
BaseRingElemType,
RingType,
RingElemType,
DomainMultSetType,
CodomainMultSetType
} <: AbsLocalizedRingHom{
RingType, RingElemType, DomainMultSetType, CodomainMultSetType
}
Homomorphisms of localizations of affine algebras
``ϕ : (𝕜[x₁,…,xₘ]/I)[S⁻¹] → (𝕜[y₁,…,yₙ]/J)[T⁻¹]``
of types `MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, DomainMultSetType}` and `MPolyQuoLocRing{BaseRingType, BaseRingElemType, RingType, RingElemType, CodomainMultSetType}`.
These are completely determined by the images of the
variables ``ϕ(xᵢ) ∈ (𝕜[y₁,…,yₙ]/J)[T⁻¹]`` so that the
constructor takes as input the triple
``((𝕜[x₁,…,xₘ]/I)[S⁻¹], (𝕜[y₁,…,yₙ]/J)[T⁻¹], [ϕ(x₁),…,ϕ(xₘ)])``.
"""
@attributes mutable struct MPolyQuoLocalizedRingHom{
DomainType<:MPolyQuoLocRing,
CodomainType<:Ring,
RestrictedMapType<:Map
} <: AbsLocalizedRingHom{
DomainType,
CodomainType,
RestrictedMapType
}
domain::DomainType
codomain::CodomainType
res::RestrictedMapType
function MPolyQuoLocalizedRingHom(
L::DomainType,
S::CodomainType,
res::RestrictedMapType;
check::Bool=true
) where {DomainType<:MPolyQuoLocRing, CodomainType<:Ring, RestrictedMapType<:Map}
R = base_ring(L)
R === domain(res) || error("restriction map is not compatible")
U = inverted_set(L)
@check begin
for f in U
is_unit(S(res(f))) || error("map is not well defined")
end
for g in gens(modulus(underlying_quotient(L)))
iszero(S(res(g))) || error("map is not well defined")
end
end
return new{DomainType, CodomainType, RestrictedMapType}(L, S, res)
end
end
### type getters
domain_type(::Type{MPolyQuoLocalizedRingHom{D, C, M}}) where {D, C, M} = D
domain_type(f::MPolyQuoLocalizedRingHom) = domain_type(typeof(f))
codomain_type(::Type{MPolyQuoLocalizedRingHom{D, C, M}}) where {D, C, M} = C
codomain_type(f::MPolyQuoLocalizedRingHom) = domain_type(typeof(f))
restricted_map_type(::Type{MPolyQuoLocalizedRingHom{D, C, M}}) where {D, C, M} = M
restricted_map_type(f::MPolyQuoLocalizedRingHom) = domain_type(typeof(f))
morphism_type(::Type{R}, ::Type{S}) where {R<:MPolyQuoLocRing, S<:Ring} = MPolyQuoLocalizedRingHom{R, S, morphism_type(base_ring_type(R), S)}