| import Mathlib.RingTheory.Spectrum.Maximal.Localization |
| import Mathlib.RingTheory.Localization.AtPrime.Basic |
|
|
| namespace OAI |
|
|
| universe uR uK |
|
|
| /-! |
| # Recovering a global coefficient from its local representations |
|
|
| The denominator-ideal argument works in any commutative algebra. It identifies |
| the intersection of maximal localizations inside a fraction field with the |
| base ring, both as subalgebra membership and through actual local elements. |
| -/ |
|
|
| namespace CirculantHadamard |
|
|
| / |
| base ring's image, then `z` itself lies in that image. -/ |
| theorem exists_base_of_local_denominators |
| {R : Type uR} {K : Type uK} [CommRing R] [CommRing K] [Algebra R K] |
| (z : K) |
| (hz : ∀ m : MaximalSpectrum R, ∃ a s : R, |
| s ∉ m.asIdeal ∧ algebraMap R K s * z = algebraMap R K a) : |
| ∃ r : R, algebraMap R K r = z := by |
| classical |
| by_contra h |
| let denom : Ideal R := |
| (1 : Submodule R K).comap (LinearMap.toSpanSingleton R K z) |
| have hdenom : (1 : R) ∉ denom := by |
| simpa [denom] using h |
| obtain ⟨m, hm, hle⟩ := |
| denom.exists_le_maximal (denom.ne_top_iff_one.mpr hdenom) |
| obtain ⟨a, s, hs, hsa⟩ := hz ⟨m, hm⟩ |
| apply hs |
| apply hle |
| change s • z ∈ (1 : Submodule R K) |
| apply Submodule.mem_one.mpr |
| exact ⟨a, by simpa only [Algebra.smul_def] using hsa.symm⟩ |
|
|
| section FractionField |
|
|
| variable {R : Type uR} {K : Type uK} [CommRing R] [IsDomain R] [Field K] |
| [Algebra R K] [IsFractionRing R K] |
|
|
| / |
| field, gives a base-ring representative. -/ |
| theorem exists_base_of_mem_all_maximal_localizations |
| {z : K} |
| (hz : ∀ m : MaximalSpectrum R, |
| z ∈ Localization.subalgebra.ofField K m.asIdeal.primeCompl |
| m.asIdeal.primeCompl_le_nonZeroDivisors) : |
| ∃ r : R, algebraMap R K r = z := by |
| have hzall : z ∈ (⨅ m : MaximalSpectrum R, |
| Localization.subalgebra.ofField K m.asIdeal.primeCompl |
| m.asIdeal.primeCompl_le_nonZeroDivisors) := |
| Algebra.mem_iInf.mpr hz |
| rw [MaximalSpectrum.iInf_localization_eq_bot R K] at hzall |
| exact Algebra.mem_bot.mp hzall |
|
|
| / |
| local ring is the image of a base-ring element. -/ |
| theorem exists_base_of_localization_images |
| {z : K} |
| (hz : ∀ m : MaximalSpectrum R, ∃ y : Localization.AtPrime m.asIdeal, |
| Localization.mapToFractionRing K m.asIdeal.primeCompl |
| (Localization.AtPrime m.asIdeal) |
| m.asIdeal.primeCompl_le_nonZeroDivisors y = z) : |
| ∃ r : R, algebraMap R K r = z := by |
| apply exists_base_of_mem_all_maximal_localizations |
| intro m |
| obtain ⟨y, hy⟩ := hz m |
| change ∃ (a s : R) (_ : s ∈ m.asIdeal.primeCompl), |
| z = algebraMap R K a * (algebraMap R K s)⁻¹ |
| exact (Localization.subalgebra.mem_range_mapToFractionRing_iff_ofField |
| K m.asIdeal.primeCompl m.asIdeal.primeCompl_le_nonZeroDivisors |
| (Localization.AtPrime m.asIdeal) z).mp ⟨y, hy⟩ |
|
|
| end FractionField |
|
|
| end CirculantHadamard |
|
|
| end OAI |
|
|
