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Localized Rings and Their Ideals

We recall the definition of localization. All rings considered are commutative, with multiplicative identity 1. Let R be a ring, and let U \subset R be a multiplicatively closed subset. That is,

$$1 \in U \;\text{ and }\; u, v \in U \;\Rightarrow \; u\cdot v \in U.$$

Consider the equivalence relation on R\times U defined by setting

$$(r,u)\sim (r', u') \;\text{ iff }\; v(r u'-u r')=0 \;{\text{ for some }}\; v\in U.$$

Write \frac{r}{u} for the equivalence class of (r, u) and R[U^{-1}] for the set of all equivalence classes. Mimicking the standard arithmetic for fractions, R[U^{-1}] can be made into a ring. This ring is called the localization of R at U. It comes equipped with the natural ring homomorphism

$$\iota : R\to R[U^{-1}],\; r \mapsto \frac{r}{1}.$$

Given an R-module M, the analogous construction yields an R[U^{-1}]-module M[U^{-1}] which is called the localization of M at U. See the section on [modules](@ref modules_multivariate).

Our focus in this section is on localizing multivariate polynomial rings and their quotients. The starting point for this is to provide functionality for handling (several types of) multiplicatively closed subsets of multivariate polynomial rings. Given such a polynomial ring R and a multiplicatively closed subset U of R whose type is supported by OSCAR, entering localization(R, U) creates the localization of R at U. Given a quotient RQ of R, with projection map p : R \to RQ, and given a multiplicatively closed subset U of R, entering localization(RQ, U) creates the localization of RQ at p(U): Since every multiplicatively closed subset of RQ is of type p(U) for some U, there is no need to support an extra type for multiplicatively closed subsets of quotients.

!!! note Most functions described here rely on the computation of standard bases. Recall that OSCAR supports standard bases for multivariate polynomial rings over fields (exact fields supported by OSCAR) and for multivariate polynomial rings over the integers.

Types

The OSCAR types discussed in this section are all parametrized. To simplify the presentation, details on the parameters are omitted.

All types for multiplicatively closed subsets of rings belong to the abstract type AbsMultSet. For multiplicatively closed subsets of multivariate polynomial rings, there are the abstract subtype AbsPolyMultSet and its concrete descendants MPolyComplementOfKPointIdeal, MPolyComplementOfPrimeIdeal, and MPolyPowersOfElement.

The general abstract type for localizations of rings is AbsLocalizedRing. For localizations of multivariate polynomial rings, there is the concrete subtype MPolyLocRing. For localizations of quotients of multivariate polynomial rings, there is the concrete subtype MPolyQuoLocRing.

Constructors

Multiplicatively Closed Subsets

In accordance with the above mentioned types, we have the following constructors for multiplicatively closed subsets of multivariate polynomial rings.

complement_of_point_ideal(R::MPolyRing, a::Vector)
complement_of_prime_ideal(P::MPolyIdeal; check::Bool=false)
powers_of_element(f::MPolyRingElem)

It is also possible to build products of multiplicatively closed sets already given:

product(T::AbsMPolyMultSet, U::AbsMPolyMultSet)

Containment in multiplicatively closed subsets can be checked via the in function:

in(f::MPolyRingElem, U::AbsMPolyMultSet)

Localized Rings

localization(R::MPolyRing, U::AbsMPolyMultSet)

localization(RQ::MPolyQuoRing, U::AbsMPolyMultSet)

Data associated to Localized Rings

If Rloc is the localization of a multivariate polynomial ring R at a multiplicatively closed subset U of R, then

  • base_ring(Rloc) refers to R, and
  • inverted_set(Rloc) to U.

If RQ is a quotient of a multivariate polynomial ring R, p : R \to RQ is the projection map, U is a multiplicatively closed subset of R, and RQL is the localization of RQ at p(U), then

  • base_ring(RQL) refers to R, and
  • inverted_set(RQL) to U.

This reflects the way of creating localizations of quotients of multivariate polynomial rings in OSCAR.

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

julia> P = ideal(R, [x])
Ideal generated by
  x

julia> U = complement_of_prime_ideal(P)
Complement
  of prime ideal (x)
  in multivariate polynomial ring in 3 variables over QQ

julia> Rloc, _ = localization(U);

julia> R === base_ring(Rloc)
true

julia> U === inverted_set(Rloc)
true

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 generated by
  2*x^2 - y^3
  2*x^2 - y^5

julia> P = ideal(R, [y-1, x-a])
Ideal generated by
  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 K

julia> RQ, _ = quo(R, I);

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

julia> R == base_ring(RQL)
true

julia> U == inverted_set(RQL)
true

Elements of Localized Rings

Types

The general abstract type for elements of localizations of rings is AbsLocalizedRingElem. For elements of localizations of multivariate polynomial rings, there is the concrete subtype MPolyLocRingElem. For elements of localizations of quotients of multivariate polynomial rings, there is the concrete subtype MPolyQuoLocRingElem.

Creating Elements of Localized Rings

If Rloc is the localization of a multivariate polynomial ring R at a multiplicatively closed subset U of R, then elements of Rloc are created as (fractions of) images of elements of R under the localization map or by coercing (pairs of) elements of R into fractions.

If RQ is a quotient of a multivariate polynomial ring R, p : R \to RQ is the projection map, U is a multiplicatively closed subset of R, and RQL is the localization of RQ at p(U), then elements of RQL are created similarly, starting from elements of R.

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> P = ideal(R, [x])
Ideal generated by
  x

julia> U = complement_of_prime_ideal(P)
Complement
  of prime ideal (x)
  in multivariate polynomial ring in 3 variables over QQ

julia> Rloc, iota = localization(U);

julia> iota(x)
x

julia> Rloc(x)
x

julia> f = iota(y)/iota(z)
y/z

julia> g = Rloc(y, z)
y/z

julia> X, Y, Z = Rloc.(gens(R));

julia> h = Y/Z
y/z

julia> f == g == h
true

julia> f+g
2*y/z

julia> f*g
y^2/z^2

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 generated by
  2*x^2 - y^3
  2*x^2 - y^5

julia> P = ideal(R, [y-1, x-a])
Ideal generated by
  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 K

julia> RQ, p = quo(R, I);

julia> RQL, iota = localization(RQ, U);

julia> phi = compose(p, iota);

julia> phi(x)
x

julia> RQL(x)
x

julia> f = phi(x)/phi(y)
x/y

julia> g = RQL(x, y)
x/y

julia> X, Y = gens(RQL);

julia> h = X/Y
x/y

julia> f == g == h
true

julia> f+g
2*x/y

julia> f*g
x^2/y^2

Data Associated to Elements of Localized Rings

If Rloc is a localization of a multivariate polynomial ring R, and f is an element of Rloc, internally represented by a pair (r, u) of elements of R, then

  • parent(f) refers to Rloc,
  • numerator(f) to r, and
  • denominator(f) to u. If RQL is a localization of a quotient RQ of a multivariate polynomial ring R, and f is an element of RQL, internally represented by a pair (r, u) of elements of R, then
  • parent(f) refers to RQL,
  • numerator(f) to the image of r in RQ, and
  • denominator(f) to the image of u in RQ. That is, the behavior of the functions numerator and denominator reflects the mathematical viewpoint of representing f by pairs of elements of RQ and not the internal representation of f as pairs of elements of R.
Examples
julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> P = ideal(R, [x])
Ideal generated by
  x

julia> U = complement_of_prime_ideal(P)
Complement
  of prime ideal (x)
  in multivariate polynomial ring in 3 variables over QQ

julia> Rloc, iota = localization(U);

julia> f = iota(x)/iota(y)
x/y

julia> parent(f)
Localization
  of multivariate polynomial ring in 3 variables x, y, z
    over rational field
  at complement of prime ideal (x)

julia> g = iota(y)/iota(z)
y/z

julia> r = numerator(f*g)
x

julia> u = denominator(f*g)
z

julia> typeof(r) == typeof(u) <: MPolyRingElem
true

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 generated by
  2*x^2 - y^3
  2*x^2 - y^5

julia> P = ideal(R, [y-1, x-a])
Ideal generated by
  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 K

julia> RQ, p = quo(R, I);

julia> RQL, iota = localization(RQ, U);

julia> phi = compose(p, iota);

julia> f = phi(x)
x

julia> parent(f)
Localization
  of quotient
    of multivariate polynomial ring in 2 variables x, y
      over number field of degree 2 over QQ
    by ideal (2*x^2 - y^3, 2*x^2 - y^5)
  at complement of prime ideal (y - 1, x - a)

julia> g = f/phi(y)
x/y

julia> r = numerator(f*g)
x^2

julia> u = denominator(f*g)
y

julia> typeof(r) == typeof(u) <: MPolyQuoRingElem
true

Tests on Elements of Localized Rings

is_unit(f::MPolyLocRingElem)

is_unit(f::MPolyQuoLocRingElem)

Homomorphisms from Localized Rings

The general abstract type for ring homomorphisms starting from localized rings is AbsLocalizedRingHom. For ring homomorphisms starting from localizations of multivariate polynomial rings, there is the concrete subtype MPolyLocalizedRingHom. For ring homomorphisms starting from quotients of multivariate polynomial rings, there is the concrete subtype MPolyQuoLocalizedRingHom. We describe the construction of such homomorphisms. Let

  • R be a multivariate polynomial ring
  • U be a multiplicatively closed subset of R,
  • RQ = R/I be a quotient of R with projection map p : R \to RQ,
  • Rloc (RQL) be the localization of R at U (of RQ at p(U)), and
  • S be another ring. Then, to give a ring homomorphism PHI from Rloc to S (fromRQL to S) is the same as to give a ring homomorphism phi from R to S which sends elements of U to units in S (and elements of I to zero). That is, PHI is determined by composing it with the localization map R \to Rloc (by composing it with the composition of the localization map RQ \to RQL and the projection map R \to RQ). The constructors below take this into account.
hom(Rloc::MPolyLocRing, S::Ring, F::Map)

Given a ring homomorphism PHI from Rloc to S (from RQL to S), domain(PHI) and codomain(PHI) refer to Rloc and S (RQL and S), respectively. The corresponding homomorphism phi from R to S is recovered as follows:

restricted_map(PHI::MPolyLocalizedRingHom)

Ideals in Localized Rings

Types

The general abstract type for ideals in localized rings is AbsLocalizedIdeal. For ideals in localizations of multivariate polynomial rings, there is the concrete subtype MPolyLocalizedIdeal. For ideals in localizations of quotients of multivariate polynomial rings, there is the concrete subtype MPolyQuoLocalizedIdeal.

Constructors

Given a localization Rloc of a multivariate polynomial ring R, and given a vector V of elements of Rloc (of R), the ideal of Rloc which is generated by (the images) of the entries of V is created by entering ideal(Rloc, V). The construction of ideals in localizations of quotients of multivariate polynomial rings is similar..

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

julia> f = x^3+y^4
x^3 + y^4

julia> V = [derivative(f, i) for i=1:2]
2-element Vector{QQMPolyRingElem}:
 3*x^2
 4*y^3

julia> U = complement_of_point_ideal(R, [0, 0]);

julia> Rloc, _ = localization(R, U);

julia> MI = ideal(Rloc, V)
Ideal generated by
  3*x^2
  4*y^3

Data Associated to Ideals

If I is an ideal of a localized multivariate polynomial ring Rloc, then

  • base_ring(I) refers to Rloc,
  • gens(I) to the generators of I,
  • number_of_generators(I) / ngens(I) to the number of these generators, and
  • gen(I, k) as well as I[k] to the k-th such generator.

Similarly, if I is an ideal of a localized quotient of a multivariate polynomial ring.

Operations on Ideals

If I, J are ideals of a localized multivariate polynomial ring Rloc, then

  • I^k refers to the k-th power of I,
  • I+J, I*J, and intersect(I, J) to the sum, product, and intersection of I and J, and
  • quotient(I, J) as well as I:J to the ideal quotient of I by J.

Similarly, if I and J are ideals of a localized quotient of a multivariate polynomial ring.

Tests on Ideals

The usual tests f in J, issubset(I, J), and I == J are available.

Saturation

If Rloc is the localization of a multivariate polynomial ring R at a multiplicative subset U of R, then the ideal theory of Rloc is a simplified version of the ideal theory of R (see, for instance, Eis95). In particular, each ideal I of Rloc is the extension $J\cdot Rloc$ of an ideal $J$ of $R$. The ideal

$${f\in R \mid uf\in J \text{ for some } u\in U}$$

is independent of the choice of $J$ and is the largest ideal of R which extends to I. It is, thus, the contraction of I to R, that is, the preimage of I under the localization map. We call this ideal the saturation of I over R. In OSCAR, it is obtained by entering saturated_ideal(I).

If RQL is the localization of a quotient RQ of a multivariate polynomial ring R, and I is an ideal of RQL, then the return value of saturated_ideal(I) is the preimage of the saturation of I over RQ under the projection map R \to RQ (and not the saturation of I over RQ itself).

saturated_ideal(I::MPolyLocalizedIdeal)