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Mathlib.RingTheory.RingHom.FiniteType

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.

theorem IsLocalization.exists_smul_mem_of_mem_adjoin {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {S' : Type u_4} [CommRing S'] [Algebra S S'] [Algebra R S'] [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S') (A : Subalgebra R S) (hA₁ : (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.

theorem IsLocalization.lift_mem_adjoin_finsetIntegerMultiple {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (M : Submonoid R) {S' : Type u_4} [CommRing S'] [Algebra S 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 (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.

theorem Algebra.FiniteType.of_span_eq_top_target {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (s : Set S) (hs : Ideal.span s = ) (h : xs, FiniteType R (Localization.Away x)) :

Finite-type can be checked on a standard covering of the target.

theorem Algebra.FiniteType.of_span_eq_top_source {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (s : Set R) (hs : Ideal.span s = ) (h : is, FiniteType (Localization.Away i) (TensorProduct R (Localization.Away i) S)) :

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

theorem RingHom.localization_away_map_finiteType (R S R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R'] (f : R →+* S) [Algebra S S'] (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : f.FiniteType) :
@[deprecated RingHom.finiteType_isLocal (since := "2025-03-01")]

Alias of RingHom.finiteType_isLocal.

@[deprecated IsLocalization.lift_mem_adjoin_finsetIntegerMultiple (since := "2025-03-14")]
theorem RingHom.IsLocalization.lift_mem_adjoin_finsetIntegerMultiple {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (M : Submonoid R) {S' : Type u_4} [CommRing S'] [Algebra S 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) :

Alias of IsLocalization.lift_mem_adjoin_finsetIntegerMultiple.


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.

@[deprecated IsLocalization.exists_smul_mem_of_mem_adjoin (since := "2025-03-14")]
theorem RingHom.IsLocalization.exists_smul_mem_of_mem_adjoin {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {S' : Type u_4} [CommRing S'] [Algebra S 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

Alias of IsLocalization.exists_smul_mem_of_mem_adjoin.


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.