Localizations of commutative rings at the complement of a prime ideal #

Main definitions #

IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*) expresses that S is a localization at (the complement of) a prime ideal P, as an abbreviation of IsLocalization P.prime_compl S

Main results #

IsLocalization.AtPrime.isLocalRing: a theorem (not an instance) stating a localization at the complement of a prime ideal is a local ring

Implementation notes #

See RingTheory.Localization.Basic for a design overview.

Tags #

localization, ring localization, commutative ring localization, characteristic predicate, commutative ring, field of fractions

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def Ideal.primeCompl {R : Type u_1} [CommSemiring R] (P : Ideal R) [hp : P.IsPrime] :

Submonoid R

The complement of a prime ideal P ⊆ R is a submonoid of R.

Equations

P.primeCompl = { carrier := (↑P)ᶜ, mul_mem' := ⋯, one_mem' := ⋯ }

Instances For

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theorem Ideal.primeCompl_le_nonZeroDivisors {R : Type u_1} [CommSemiring R] (P : Ideal R) [hp : P.IsPrime] [NoZeroDivisors R] :

P.primeCompl ≤ nonZeroDivisors R

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@[reducible, inline]

abbrev IsLocalization.AtPrime {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (P : Ideal R) [hp : P.IsPrime] :

Prop

Given a prime ideal P, the typeclass IsLocalization.AtPrime S P states that S is isomorphic to the localization of R at the complement of P.

Equations

IsLocalization.AtPrime S P = IsLocalization P.primeCompl S

Instances For

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@[reducible, inline]

abbrev Localization.AtPrime {R : Type u_1} [CommSemiring R] (P : Ideal R) [hp : P.IsPrime] :

Type u_1

Given a prime ideal P, Localization.AtPrime P is a localization of R at the complement of P, as a quotient type.

Equations

Localization.AtPrime P = Localization P.primeCompl

Instances For

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theorem IsLocalization.AtPrime.Nontrivial {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (P : Ideal R) [hp : P.IsPrime] [IsLocalization.AtPrime S P] :

Nontrivial S

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theorem IsLocalization.AtPrime.isLocalRing {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (P : Ideal R) [hp : P.IsPrime] [IsLocalization.AtPrime S P] :

IsLocalRing S

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@[deprecated IsLocalization.AtPrime.isLocalRing]

theorem IsLocalization.AtPrime.localRing {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (P : Ideal R) [hp : P.IsPrime] [IsLocalization.AtPrime S P] :

IsLocalRing S

Alias of IsLocalization.AtPrime.isLocalRing.

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instance Localization.AtPrime.isLocalRing {R : Type u_1} [CommSemiring R] (P : Ideal R) [hp : P.IsPrime] :

IsLocalRing (Localization P.primeCompl)

The localization of R at the complement of a prime ideal is a local ring.

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instance IsLocalization.isDomain_of_local_atPrime {A : Type u_4} [CommRing A] [IsDomain A] {P : Ideal A} (x✝ : P.IsPrime) :

IsDomain (Localization.AtPrime P)

The localization of an integral domain at the complement of a prime ideal is an integral domain.

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def IsLocalization.AtPrime.orderIsoOfPrime {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] :

{ p : Ideal S // p.IsPrime } ≃o { p : Ideal R // p.IsPrime ∧ p ≤ I }

The prime ideals in the localization of a commutative ring at a prime ideal I are in order-preserving bijection with the prime ideals contained in I.

Equations

One or more equations did not get rendered due to their size.

Instances For

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theorem IsLocalization.AtPrime.isUnit_to_map_iff {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] (x : R) :

IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl

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theorem IsLocalization.AtPrime.to_map_mem_maximal_iff {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] (x : R) (h : IsLocalRing S := ⋯) :

(algebraMap R S) x ∈ IsLocalRing.maximalIdeal S ↔ x ∈ I

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theorem IsLocalization.AtPrime.comap_maximalIdeal {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] (h : IsLocalRing S := ⋯) :

Ideal.comap (algebraMap R S) (IsLocalRing.maximalIdeal S) = I

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theorem IsLocalization.AtPrime.isUnit_mk'_iff {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] (x : R) (y : ↥I.primeCompl) :

IsUnit (IsLocalization.mk' S x y) ↔ x ∈ I.primeCompl

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theorem IsLocalization.AtPrime.mk'_mem_maximal_iff {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I] (x : R) (y : ↥I.primeCompl) (h : IsLocalRing S := ⋯) :

IsLocalization.mk' S x y ∈ IsLocalRing.maximalIdeal S ↔ x ∈ I

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theorem Localization.AtPrime.comap_maximalIdeal {R : Type u_1} [CommSemiring R] {I : Ideal R} [hI : I.IsPrime] :

Ideal.comap (algebraMap R (Localization.AtPrime I)) (IsLocalRing.maximalIdeal (Localization I.primeCompl)) = I

The unique maximal ideal of the localization at I.primeCompl lies over the ideal I.

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theorem Localization.AtPrime.map_eq_maximalIdeal {R : Type u_1} [CommSemiring R] {I : Ideal R} [hI : I.IsPrime] :

Ideal.map (algebraMap R (Localization.AtPrime I)) I = IsLocalRing.maximalIdeal (Localization I.primeCompl)

The image of I in the localization at I.primeCompl is a maximal ideal, and in particular it is the unique maximal ideal given by the local ring structure AtPrime.isLocalRing

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theorem Localization.le_comap_primeCompl_iff {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] {I : Ideal R} [hI : I.IsPrime] {J : Ideal P} [J.IsPrime] {f : R →+* P} :

I.primeCompl ≤ Submonoid.comap f J.primeCompl ↔ Ideal.comap f J ≤ I

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noncomputable def Localization.localRingHom {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) :

Localization.AtPrime I →+* Localization.AtPrime J

For a ring hom f : R →+* S and a prime ideal J in S, the induced ring hom from the localization of R at J.comap f to the localization of S at J.

To make this definition more flexible, we allow any ideal I of R as input, together with a proof that I = J.comap f. This can be useful when I is not definitionally equal to J.comap f.

Equations

Localization.localRingHom I J f hIJ = IsLocalization.map (Localization.AtPrime J) f ⋯

Instances For

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theorem Localization.localRingHom_to_map {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) (x : R) :

(Localization.localRingHom I J f hIJ) ((algebraMap R (Localization.AtPrime I)) x) = (algebraMap P (Localization.AtPrime J)) (f x)

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theorem Localization.localRingHom_mk' {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) (x : R) (y : ↥I.primeCompl) :

(Localization.localRingHom I J f hIJ) (IsLocalization.mk' (Localization.AtPrime I) x y) = IsLocalization.mk' (Localization.AtPrime J) (f x) ⟨f ↑y, ⋯⟩

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instance Localization.isLocalHom_localRingHom {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) :

IsLocalHom (Localization.localRingHom I J f hIJ)

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@[deprecated Localization.isLocalHom_localRingHom]

theorem Localization.isLocalRingHom_localRingHom {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) :

IsLocalHom (Localization.localRingHom I J f hIJ)

Alias of Localization.isLocalHom_localRingHom.

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theorem Localization.localRingHom_unique {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) {j : Localization.AtPrime I →+* Localization.AtPrime J} (hj : ∀ (x : R), j ((algebraMap R (Localization.AtPrime I)) x) = (algebraMap P (Localization.AtPrime J)) (f x)) :

Localization.localRingHom I J f hIJ = j

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@[simp]

theorem Localization.localRingHom_id {R : Type u_1} [CommSemiring R] (I : Ideal R) [hI : I.IsPrime] :

Localization.localRingHom I I (RingHom.id R) ⋯ = RingHom.id (Localization.AtPrime I)

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theorem Localization.localRingHom_comp {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] {S : Type u_4} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P) [hK : K.IsPrime] (f : R →+* S) (hIJ : I = Ideal.comap f J) (g : S →+* P) (hJK : J = Ideal.comap g K) :

Localization.localRingHom I K (g.comp f) ⋯ = (Localization.localRingHom J K g hJK).comp (Localization.localRingHom I J f hIJ)

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@[reducible, inline]

noncomputable abbrev Localization.AtPrime.mapPiEvalRingHom {ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] :

Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I) →+* Localization.AtPrime I

Localization.localRingHom specialized to a projection homomorphism from a product ring.

Equations

Localization.AtPrime.mapPiEvalRingHom I = Localization.localRingHom (Ideal.comap (Pi.evalRingHom R i) I) I (Pi.evalRingHom R i) ⋯

Instances For

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theorem Localization.AtPrime.mapPiEvalRingHom_bijective {ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] :

Function.Bijective ⇑(Localization.AtPrime.mapPiEvalRingHom I)

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theorem Localization.AtPrime.mapPiEvalRingHom_comp_algebraMap {ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] :

(Localization.AtPrime.mapPiEvalRingHom I).comp (algebraMap ((i : ι) → R i) (Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I))) = (algebraMap (R i) (Localization.AtPrime I)).comp (Pi.evalRingHom R i)

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theorem Localization.AtPrime.mapPiEvalRingHom_algebraMap_apply {ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] {r : (i : ι) → R i} :

(Localization.AtPrime.mapPiEvalRingHom I) ((algebraMap ((i : ι) → R i) (Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I))) r) = (algebraMap (R i) (Localization.AtPrime I)) (r i)

Documentation

Mathlib.RingTheory.Localization.AtPrime

Localizations of commutative rings at the complement of a prime ideal #

Main definitions #

Main results #

Implementation notes #

Tags #