The meta properties of finite-type ring homomorphisms. #

Main results #

Let R be a commutative ring, S is an R-algebra, M be a submonoid of R.

finiteType_localizationPreserves : If S is a finite type R-algebra, then S' = M⁻¹S is a finite type R' = M⁻¹R-algebra.
finiteType_ofLocalizationSpan : S is a finite type R-algebra if there exists a set { r } that spans R such that Sᵣ is a finite type Rᵣ-algebra. *RingHom.finiteType_is_local: RingHom.FiniteType is a local property.

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theorem RingHom.finiteType_stableUnderComposition :

RingHom.StableUnderComposition @RingHom.FiniteType

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theorem RingHom.finiteType_localizationPreserves :

RingHom.LocalizationPreserves @RingHom.FiniteType

If S is a finite type R-algebra, then S' = M⁻¹S is a finite type R' = M⁻¹R-algebra.

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theorem RingHom.localization_away_map_finiteType {R : Type u} {S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (R' : Type u) (S' : Type u) [CommRing R'] [CommRing S'] [Algebra R R'] [Algebra S S'] (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : f.FiniteType) :

(IsLocalization.Away.map R' S' f r).FiniteType

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theorem RingHom.IsLocalization.exists_smul_mem_of_mem_adjoin {R : Type u} {S : Type u} [CommRing R] [CommRing S] {S' : Type u} [CommRing S'] [Algebra S S'] [Algebra R S] [Algebra R S'] [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S') (A : Subalgebra R S) (hA₁ : ↑(IsLocalization.finsetIntegerMultiple M s) ⊆ ↑A) (hA₂ : M ≤ A.toSubmonoid) (hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s) :

∃ (m : ↥M), m • x ∈ A

Let S be an R-algebra, M a submonoid of S, S' = M⁻¹S. Suppose the image of some x : S falls in the adjoin of some finite s ⊆ S' over R, and A is an R-subalgebra of S containing both M and the numerators of s. Then, there exists some m : M such that m • x falls in A.

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theorem RingHom.IsLocalization.lift_mem_adjoin_finsetIntegerMultiple {R : Type u} {S : Type u} [CommRing R] [CommRing S] (M : Submonoid R) {S' : Type u} [CommRing S'] [Algebra S S'] [Algebra R S] [Algebra R S'] [IsScalarTower R S S'] [IsLocalization (Submonoid.map (algebraMap R S) M) S'] (x : S) (s : Finset S') (hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s) :

∃ (m : ↥M), m • x ∈ Algebra.adjoin R ↑(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)

Let S be an R-algebra, M a submonoid of R, and S' = M⁻¹S. If the image of some x : S falls in the adjoin of some finite s ⊆ S' over R, then there exists some m : M such that m • x falls in the adjoin of IsLocalization.finsetIntegerMultiple _ s over R.