Ultra Product exists ring Equiv of bdd Above card
UltraProduct.exists_ringEquiv_of_bddAbove_card
Plain-language statement
Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous ring homs f i : R →+* R i, there exists F-many i such that 𝒰(Rᵢ) ≃+* R i and this map is compatible with f.
Exact Lean statement
lemma UltraProduct.exists_ringEquiv_of_bddAbove_card
[TopologicalSpace R₀]
[IsTopologicalRing R₀]
[Algebra.TopologicallyFG ℤ R₀]
[∀ i, TopologicalSpace (R i)]
[∀ i, IsTopologicalRing (R i)]
[∀ i, T2Space (R i)] (F : Ultrafilter ι)
(N : ℕ) (H : ∀ᶠ i in F, Finite (R i) ∧ Nat.card (R i) < N)
(f : ∀ i, R₀ →+* R i) (hf : ∀ᶠ i in F, Continuous (f i)) :
∀ᶠ i in F, ∃ e : UltraProduct R F ≃+*
R i, e.toRingHom.comp ((π R F).comp (RingHom.pi f)) = f iFormal artifact
Lean source
lemma UltraProduct.exists_ringEquiv_of_bddAbove_card [TopologicalSpace R₀] [IsTopologicalRing R₀] [Algebra.TopologicallyFG ℤ R₀] [∀ i, TopologicalSpace (R i)] [∀ i, IsTopologicalRing (R i)] [∀ i, T2Space (R i)] (F : Ultrafilter ι) (N : ℕ) (H : ∀ᶠ i in F, Finite (R i) ∧ Nat.card (R i) < N) (f : ∀ i, R₀ →+* R i) (hf : ∀ᶠ i in F, Continuous (f i)) : ∀ᶠ i in F, ∃ e : UltraProduct R F ≃+* R i, e.toRingHom.comp ((π R F).comp (RingHom.pi f)) = f i := by classical letI := fun i ↦ (f i).toAlgebra have := UltraProduct.exists_algEquiv_of_bddAbove_card (R₀ := R₀) F N H (by filter_upwards [hf] with i hi; exact ⟨show Continuous fun p : R₀ × R i ↦ f i p.1 * p.2 by continuity⟩) filter_upwards [this] with i ⟨e⟩ exact ⟨e, e.toAlgHom.comp_algebraMap⟩- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/Patching/Ultraproduct.lean:371-388
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