Localizations away from an element #

Main definitions #

IsLocalization.Away (x : R) S expresses that S is a localization away from x, as an abbreviation of IsLocalization (Submonoid.powers x) S.
exists_reduced_fraction' (hb : b ≠ 0) produces a reduced fraction of the form b = a * x^n for some n : ℤ and some a : R that is not divisible by x.

Implementation notes #

See Mathlib/RingTheory/Localization/Basic.lean for a design overview.

Tags #

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

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@[reducible, inline]

abbrev IsLocalization.Away {R : Type u_1} [CommSemiring R] (x : R) (S : Type u_4) [CommSemiring S] [Algebra R S] :

Prop

Given x : R, the typeclass IsLocalization.Away x S states that S is isomorphic to the localization of R at the submonoid generated by x. See IsLocalization.Away.mk for a specialized constructor.

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noncomputable def IsLocalization.Away.invSelf {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] :

Given x : R and a localization map F : R →+* S away from x, invSelf is (F x)⁻¹.

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@[simp]

theorem IsLocalization.Away.mul_invSelf {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] :

(algebraMap R S) x * invSelf x = 1

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noncomputable def IsLocalization.Away.sec {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] (s : S) :

R × ℕ

For s : S with S being the localization of R away from x, this is a choice of (r, n) : R × ℕ such that s * algebraMap R S (x ^ n) = algebraMap R S r.

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theorem IsLocalization.Away.sec_spec {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] (s : S) :

s * (algebraMap R S) (x ^ (sec x s).2) = (algebraMap R S) (sec x s).1

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theorem IsLocalization.Away.algebraMap_pow_isUnit {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] (n : ℕ) :

IsUnit ((algebraMap R S) x ^ n)

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theorem IsLocalization.Away.algebraMap_isUnit {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] :

IsUnit ((algebraMap R S) x)

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theorem IsLocalization.Away.algebraMap_isUnit_iff {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] {y : R} :

IsUnit ((algebraMap R S) y) ↔ ∃ (n : ℕ), y ∣ x ^ n

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theorem IsLocalization.Away.surj {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] (z : S) :

∃ (n : ℕ) (a : R), z * (algebraMap R S) x ^ n = (algebraMap R S) a

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theorem IsLocalization.Away.exists_of_eq {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] {a b : R} (h : (algebraMap R S) a = (algebraMap R S) b) :

∃ (n : ℕ), x ^ n * a = x ^ n * b

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theorem IsLocalization.Away.mk {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (r : R) (map_unit : IsUnit ((algebraMap R S) r)) (surj : ∀ (s : S), ∃ (n : ℕ) (a : R), s * (algebraMap R S) r ^ n = (algebraMap R S) a) (exists_of_eq : ∀ (a b : R), (algebraMap R S) a = (algebraMap R S) b → ∃ (n : ℕ), r ^ n * a = r ^ n * b) :

Away r S

Specialized constructor for IsLocalization.Away.

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theorem IsLocalization.Away.of_associated {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {r r' : R} (h : Associated r r') [Away r S] :

Away r' S

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theorem IsLocalization.Away.iff_of_associated {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {r r' : R} (h : Associated r r') :

Away r S ↔ Away r' S

If r and r' are associated elements of R, an R-algebra S is the localization of R away from r if and only of it is the localization of R away from r'.

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theorem IsLocalization.Away.isUnit_of_dvd {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] (x : R) [Away x S] {r : R} (h : r ∣ x) :

IsUnit ((algebraMap R S) r)

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noncomputable def IsLocalization.Away.lift {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (x : R) [Away x S] {g : R →+* P} (hg : IsUnit (g x)) :

S →+* P

Given x : R, a localization map F : R →+* S away from x, and a map of CommSemirings g : R →+* P such that g x is invertible, the homomorphism induced from S to P sending z : S to g y * (g x)⁻ⁿ, where y : R, n : ℕ are such that z = F y * (F x)⁻ⁿ.

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@[simp]

theorem IsLocalization.Away.lift_eq {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (x : R) [Away x S] {g : R →+* P} (hg : IsUnit (g x)) (a : R) :

(lift x hg) ((algebraMap R S) a) = g a

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@[simp]

theorem IsLocalization.Away.lift_comp {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (x : R) [Away x S] {g : R →+* P} (hg : IsUnit (g x)) :

(lift x hg).comp (algebraMap R S) = g

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noncomputable def IsLocalization.Away.awayToAwayLeft {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (x : R) [Away x S] (y : R) [Algebra R P] [Away (y * x) P] :

S →+* P

Given x y : R and localizations S, P away from x and y * x respectively, the homomorphism induced from S to P.

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noncomputable def IsLocalization.Away.awayToAwayRight {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (x : R) [Away x S] (y : R) [Algebra R P] [Away (x * y) P] :

S →+* P

Given x y : R and localizations S, P away from x and x * y respectively, the homomorphism induced from S to P.

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theorem IsLocalization.Away.awayToAwayLeft_eq {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (x : R) [Away x S] (y : R) [Algebra R P] [Away (y * x) P] (a : R) :

(awayToAwayLeft x y) ((algebraMap R S) a) = (algebraMap R P) a

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theorem IsLocalization.Away.awayToAwayRight_eq {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (x : R) [Away x S] (y : R) [Algebra R P] [Away (x * y) P] (a : R) :

(awayToAwayRight x y) ((algebraMap R S) a) = (algebraMap R P) a

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noncomputable def IsLocalization.Away.map {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (Q : Type u_4) [CommSemiring Q] [Algebra P Q] (f : R →+* P) (r : R) [Away r S] [Away (f r) Q] :

S →+* Q

Given a map f : R →+* S and an element r : R, we may construct a map Rᵣ →+* Sᵣ.

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instance IsLocalization.Away.instMapRingHomPowersOfCoe {A : Type u_5} [CommSemiring A] {B : Type u_6} [CommSemiring B] (Bₚ : Type u_8) [CommSemiring Bₚ] [Algebra B Bₚ] {f : A →+* B} (a : A) [Away (f a) Bₚ] :

IsLocalization (Submonoid.map f (Submonoid.powers a)) Bₚ

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noncomputable def IsLocalization.Away.mapₐ {R : Type u_1} [CommSemiring R] {A : Type u_5} [CommSemiring A] [Algebra R A] {B : Type u_6} [CommSemiring B] [Algebra R B] (Aₚ : Type u_7) [CommSemiring Aₚ] [Algebra A Aₚ] [Algebra R Aₚ] [IsScalarTower R A Aₚ] (Bₚ : Type u_8) [CommSemiring Bₚ] [Algebra B Bₚ] [Algebra R Bₚ] [IsScalarTower R B Bₚ] (f : A →ₐ[R] B) (a : A) [Away a Aₚ] [Away (f a) Bₚ] :

Aₚ →ₐ[R] Bₚ

Given a algebra map f : A →ₐ[R] B and an element a : A, we may construct a map Aₐ →ₐ[R] Bₐ.

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@[simp]

theorem IsLocalization.Away.mapₐ_apply {R : Type u_1} [CommSemiring R] {A : Type u_5} [CommSemiring A] [Algebra R A] {B : Type u_6} [CommSemiring B] [Algebra R B] (Aₚ : Type u_7) [CommSemiring Aₚ] [Algebra A Aₚ] [Algebra R Aₚ] [IsScalarTower R A Aₚ] (Bₚ : Type u_8) [CommSemiring Bₚ] [Algebra B Bₚ] [Algebra R Bₚ] [IsScalarTower R B Bₚ] (f : A →ₐ[R] B) (a : A) [Away a Aₚ] [Away (f a) Bₚ] (x : Aₚ) :

(mapₐ Aₚ Bₚ f a) x = (map Aₚ Bₚ f.toRingHom a) x

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theorem IsLocalization.Away.mapₐ_injective_of_injective {R : Type u_1} [CommSemiring R] {A : Type u_5} [CommSemiring A] [Algebra R A] {B : Type u_6} [CommSemiring B] [Algebra R B] {Aₚ : Type u_7} [CommSemiring Aₚ] [Algebra A Aₚ] [Algebra R Aₚ] [IsScalarTower R A Aₚ] {Bₚ : Type u_8} [CommSemiring Bₚ] [Algebra B Bₚ] [Algebra R Bₚ] [IsScalarTower R B Bₚ] {f : A →ₐ[R] B} (a : A) [Away a Aₚ] [Away (f a) Bₚ] (hf : Function.Injective ⇑f) :

Function.Injective ⇑(mapₐ Aₚ Bₚ f a)

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theorem IsLocalization.Away.mapₐ_surjective_of_surjective {R : Type u_1} [CommSemiring R] {A : Type u_5} [CommSemiring A] [Algebra R A] {B : Type u_6} [CommSemiring B] [Algebra R B] {Aₚ : Type u_7} [CommSemiring Aₚ] [Algebra A Aₚ] [Algebra R Aₚ] [IsScalarTower R A Aₚ] {Bₚ : Type u_8} [CommSemiring Bₚ] [Algebra B Bₚ] [Algebra R Bₚ] [IsScalarTower R B Bₚ] {f : A →ₐ[R] B} (a : A) [Away a Aₚ] [Away (f a) Bₚ] (hf : Function.Surjective ⇑f) :

Function.Surjective ⇑(mapₐ Aₚ Bₚ f a)

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theorem IsLocalization.Away.mul {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (T : Type u_5) [CommSemiring T] [Algebra S T] [Algebra R T] [IsScalarTower R S T] (x y : R) [Away x S] [Away ((algebraMap R S) y) T] :

Away (y * x) T

Localizing the localization of R at x at the image of y is the same as localizing R at y * x. See IsLocalization.Away.mul' for the x * y version.

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theorem IsLocalization.Away.mul' {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (T : Type u_5) [CommSemiring T] [Algebra S T] [Algebra R T] [IsScalarTower R S T] (x y : R) [Away x S] [Away ((algebraMap R S) y) T] :

Away (x * y) T

Localizing the localization of R at x at the image of y is the same as localizing R at x * y. See IsLocalization.Away.mul for the y * x version.

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instance IsLocalization.Away.instHMulAwayCoeRingHomAlgebraMap {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (x y : R) [Away x S] :

Away (y * x) (Localization.Away ((algebraMap R S) y))

Localizing the localization of R at x at the image of y is the same as localizing R at y * x.

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instance IsLocalization.Away.instHMulAwayCoeRingHomAlgebraMap_1 {R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (x y : R) [Away x S] :

Away (x * y) (Localization.Away ((algebraMap R S) y))

Localizing the localization of R at x at the image of y is the same as localizing R at x * y.

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theorem IsLocalization.Away.commutes {R : Type u_5} [CommSemiring R] (S₁ : Type u_6) (S₂ : Type u_7) (T : Type u_8) [CommSemiring S₁] [CommSemiring S₂] [CommSemiring T] [Algebra R S₁] [Algebra R S₂] [Algebra R T] [Algebra S₁ T] [Algebra S₂ T] [IsScalarTower R S₁ T] [IsScalarTower R S₂ T] (x y : R) [Away x S₁] [Away y S₂] [Away ((algebraMap R S₂) x) T] :

Away ((algebraMap R S₁) y) T

If S₁ is the localization of R away from f and S₂ is the localization away from g, then any localization T of S₂ away from f is also a localization of S₁ away from g.

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noncomputable def IsLocalization.atUnit (R : Type u_1) [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (x : R) (e : IsUnit x) [Away x S] :

R ≃ₐ[R] S

The localization away from a unit is isomorphic to the ring.

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noncomputable def IsLocalization.atOne (R : Type u_1) [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] [Away 1 S] :

R ≃ₐ[R] S

The localization at one is isomorphic to the ring.

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theorem IsLocalization.away_of_isUnit_of_bijective {R : Type u_4} (S : Type u_5) [CommSemiring R] [CommSemiring S] [Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective ⇑(algebraMap R S)) :

Away r S

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theorem IsLocalization.away_of_isIdempotentElem_of_mul {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] [Algebra R S] {e : R} (he : IsIdempotentElem e) (H : ∀ (x y : R), (algebraMap R S) x = (algebraMap R S) y ↔ e * x = e * y) (H' : Function.Surjective ⇑(algebraMap R S)) :

Away e S

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theorem IsLocalization.away_of_isIdempotentElem {R : Type u_4} {S : Type u_5} [CommRing R] [CommRing S] [Algebra R S] {e : R} (he : IsIdempotentElem e) (H : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}) (H' : Function.Surjective ⇑(algebraMap R S)) :

Away e S

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instance IsLocalization.away_fst {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] :

Away (1, 0) R

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instance IsLocalization.away_snd {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] :

Away (0, 1) S

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@[reducible, inline]

noncomputable abbrev Localization.awayLift {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (f : R →+* P) (r : R) (hr : IsUnit (f r)) :

Away r →+* P

Given a map f : R →+* S and an element r : R, such that f r is invertible, we may construct a map Rᵣ →+* S.

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theorem Localization.awayLift_mk {R : Type u_1} [CommSemiring R] {A : Type u_4} [CommSemiring A] (f : R →+* A) (r a : R) (v : A) (hv : f r * v = 1) (j : ℕ) :

(awayLift f r ⋯) (mk a ⟨r ^ j, ⋯⟩) = f a * v ^ j

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@[reducible, inline]

noncomputable abbrev Localization.awayMap {R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (f : R →+* P) (r : R) :

Away r →+* Away (f r)

Given a map f : R →+* S and an element r : R, we may construct a map Rᵣ →+* Sᵣ.

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@[reducible, inline]

noncomputable abbrev Localization.awayMapₐ {R : Type u_1} [CommSemiring R] {A : Type u_4} [CommSemiring A] [Algebra R A] {B : Type u_5} [CommSemiring B] [Algebra R B] (f : A →ₐ[R] B) (a : A) :

Away a →ₐ[R] Away (f a)

Given a map f : A →ₐ[R] B and an element a : A, we may construct a map Aₐ →ₐ[R] Bₐ.

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theorem Localization.algebraMap_injective_of_span_eq_top {R : Type u_1} [CommSemiring R] (s : Set R) (span_eq : Ideal.span s = ⊤) :

Function.Injective ⇑(algebraMap R ((a : ↑s) → Away ↑a))

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theorem Localization.existsUnique_algebraMap_eq_of_span_eq_top {R : Type u_1} [CommSemiring R] (s : Set R) (span_eq : Ideal.span s = ⊤) (f : (a : ↑s) → Away ↑a) (h : ∀ (a b : ↑s), (IsLocalization.Away.awayToAwayRight ↑a ↑b) (f a) = (IsLocalization.Away.awayToAwayLeft ↑b ↑a) (f b)) :

∃! r : R, ∀ (a : ↑s), (algebraMap R (Away ↑a)) r = f a

The sheaf condition for the structure sheaf on Spec R for a covering of the whole prime spectrum by basic opens.

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noncomputable def selfZPow {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] (m : ℤ) :

selfZPow x (m : ℤ) is x ^ m as an element of the localization away from x.

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theorem selfZPow_of_nonneg {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] {n : ℤ} (hn : 0 ≤ n) :

selfZPow x B n = (algebraMap R B) x ^ n.natAbs

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@[simp]

theorem selfZPow_natCast {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] (d : ℕ) :

selfZPow x B ↑d = (algebraMap R B) x ^ d

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@[simp]

theorem selfZPow_zero {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] :

selfZPow x B 0 = 1

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theorem selfZPow_of_neg {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] {n : ℤ} (hn : n < 0) :

selfZPow x B n = IsLocalization.mk' B 1 (Submonoid.pow x n.natAbs)

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theorem selfZPow_of_nonpos {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] {n : ℤ} (hn : n ≤ 0) :

selfZPow x B n = IsLocalization.mk' B 1 (Submonoid.pow x n.natAbs)

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@[simp]

theorem selfZPow_neg_natCast {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] (d : ℕ) :

selfZPow x B (-↑d) = IsLocalization.mk' B 1 (Submonoid.pow x d)

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@[simp]

theorem selfZPow_sub_natCast {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] {n m : ℕ} :

selfZPow x B (↑n - ↑m) = IsLocalization.mk' B (x ^ n) (Submonoid.pow x m)

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@[simp]

theorem selfZPow_add {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] {n m : ℤ} :

selfZPow x B (n + m) = selfZPow x B n * selfZPow x B m

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theorem selfZPow_mul_neg {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] (d : ℤ) :

selfZPow x B d * selfZPow x B (-d) = 1

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theorem selfZPow_neg_mul {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] (d : ℤ) :

selfZPow x B (-d) * selfZPow x B d = 1

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theorem selfZPow_pow_sub {R : Type u_1} [CommSemiring R] (x : R) (B : Type u_2) [CommSemiring B] [Algebra R B] [IsLocalization.Away x B] (a : R) (b : B) (m d : ℤ) :

selfZPow x B (m - d) * IsLocalization.mk' B a 1 = b ↔ selfZPow x B m * IsLocalization.mk' B a 1 = selfZPow x B d * b

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theorem exists_reduced_fraction' {R : Type u_3} [CommRing R] (x : R) (B : Type u_4) [CommRing B] [Algebra R B] [IsLocalization.Away x B] [IsDomain R] [WfDvdMonoid R] {b : B} (hb : b ≠ 0) (hx : Irreducible x) :

∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b

Documentation

Mathlib.RingTheory.Localization.Away.Basic

Localizations away from an element #

Main definitions #

Implementation notes #

Tags #