File size: 3,089 Bytes
cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | 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
/-- If every maximal ideal admits a denominator outside it taking `z` into the
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]
/-- Membership in every maximal localization, realized inside a common fraction
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
/-- A fraction-field element represented by an actual element of each maximal
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
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