Integer elements of a localization #

Main definitions #

IsLocalization.IsInteger is a predicate stating that x : S is in the image of R

Implementation notes #

See 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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def IsLocalization.IsInteger (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] (a : S) :

Prop

Given a : S, S a localization of R, IsInteger R a iff a is in the image of the localization map from R to S.

Equations

IsLocalization.IsInteger R a = (a ∈ RingHom.range (algebraMap R S))

Instances For

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theorem IsLocalization.isInteger_zero {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] :

IsLocalization.IsInteger R 0

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theorem IsLocalization.isInteger_one {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] :

IsLocalization.IsInteger R 1

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theorem IsLocalization.isInteger_add {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {a : S} {b : S} (ha : IsLocalization.IsInteger R a) (hb : IsLocalization.IsInteger R b) :

IsLocalization.IsInteger R (a + b)

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theorem IsLocalization.isInteger_mul {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {a : S} {b : S} (ha : IsLocalization.IsInteger R a) (hb : IsLocalization.IsInteger R b) :

IsLocalization.IsInteger R (a * b)

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theorem IsLocalization.isInteger_smul {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {a : R} {b : S} (hb : IsLocalization.IsInteger R b) :

IsLocalization.IsInteger R (a • b)

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theorem IsLocalization.exists_integer_multiple' {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (a : S) :

∃ b, IsLocalization.IsInteger R (a * ↑(algebraMap R S) ↑b)

Each element a : S has an M-multiple which is an integer.

This version multiplies a on the right, matching the argument order in LocalizationMap.surj.

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theorem IsLocalization.exists_integer_multiple {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (a : S) :

∃ b, IsLocalization.IsInteger R (↑b • a)

Each element a : S has an M-multiple which is an integer.

This version multiplies a on the left, matching the argument order in the SMul instance.

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theorem IsLocalization.exist_integer_multiples {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ι → S) :

∃ b, ∀ (i : ι), i ∈ s → IsLocalization.IsInteger R (↑b • f i)

We can clear the denominators of a Finset-indexed family of fractions.

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theorem IsLocalization.exist_integer_multiples_of_finite {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} [Finite ι] (f : ι → S) :

∃ b, ∀ (i : ι), IsLocalization.IsInteger R (↑b • f i)

We can clear the denominators of a finite indexed family of fractions.

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theorem IsLocalization.exist_integer_multiples_of_finset {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (s : Finset S) :

∃ b, ∀ (a : S), a ∈ s → IsLocalization.IsInteger R (↑b • a)

We can clear the denominators of a finite set of fractions.

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noncomputable def IsLocalization.commonDenom {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ι → S) :

{ x // x ∈ M }

A choice of a common multiple of the denominators of a Finset-indexed family of fractions.

Equations

IsLocalization.commonDenom M s f = Exists.choose (_ : ∃ b, ∀ (i : ι), i ∈ s → IsLocalization.IsInteger R (↑b • f i))

Instances For

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noncomputable def IsLocalization.integerMultiple {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ι → S) (i : { x // x ∈ s }) :

The numerator of a fraction after clearing the denominators of a Finset-indexed family of fractions.

Equations

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

Instances For

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@[simp]

theorem IsLocalization.map_integerMultiple {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ι → S) (i : { x // x ∈ s }) :

↑(algebraMap R S) (IsLocalization.integerMultiple M s f i) = IsLocalization.commonDenom M s f • f ↑i

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noncomputable def IsLocalization.commonDenomOfFinset {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] (s : Finset S) :

{ x // x ∈ M }

A choice of a common multiple of the denominators of a finite set of fractions.

Equations

IsLocalization.commonDenomOfFinset M s = IsLocalization.commonDenom M s id

Instances For

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noncomputable def IsLocalization.finsetIntegerMultiple {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] [DecidableEq R] (s : Finset S) :

Finset R

The finset of numerators after clearing the denominators of a finite set of fractions.

Equations

IsLocalization.finsetIntegerMultiple M s = Finset.image (fun t => IsLocalization.integerMultiple M s id t) (Finset.attach s)

Instances For

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theorem IsLocalization.finsetIntegerMultiple_image {R : Type u_1} [CommRing R] (M : Submonoid R) {S : Type u_2} [CommRing S] [Algebra R S] [IsLocalization M S] [DecidableEq R] (s : Finset S) :

↑(algebraMap R S) '' ↑(IsLocalization.finsetIntegerMultiple M s) = IsLocalization.commonDenomOfFinset M s • ↑s