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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 85 86 87 88 89 90 91 92 93 94 | import OAI.LinearAlgebra.CirculantHadamard.LocalDVR
import Mathlib.RingTheory.Localization.AtPrime.Basic
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Int.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.Ring.Int.Parity
namespace OAI
/-!
# The integers localized at the prime two
The ring is the actual localization of `ℤ` at the complement of `(2)`. Its
maximal ideal is generated by the image of two, and its nonzero generator makes
it a discrete valuation ring. The characteristic-zero and odd-unit interfaces
refer to this concrete localization, without assuming any valuation facts.
-/
noncomputable section
namespace CirculantHadamard.IntegerTwoLocal
/-- The prime ideal `(2)` in the integers. -/
def ideal : Ideal ℤ := Ideal.span {(2 : ℤ)}
instance ideal_isPrime : ideal.IsPrime := by
apply (Ideal.span_singleton_prime (by norm_num : (2 : ℤ) ≠ 0)).mpr
exact Nat.prime_iff_prime_int.mp Nat.prime_two
/-- The actual ring `ℤ_(2)`. -/
abbrev Ring := Localization.AtPrime ideal
/-- The localization has no zero divisors. -/
theorem isDomain : IsDomain Ring := inferInstance
/-- The complement-of-prime localization is local. -/
theorem isLocalRing : IsLocalRing Ring := inferInstance
/-- Integer coefficients embed faithfully in the concrete localization. -/
theorem algebraMap_injective : Function.Injective (algebraMap ℤ Ring) :=
IsLocalization.injective Ring ideal.primeCompl_le_nonZeroDivisors
instance charZero : CharZero Ring :=
charZero_of_injective_algebraMap algebraMap_injective
/-- Noetherianity is inherited from the integer ring by localization. -/
instance isNoetherianRing : IsNoetherianRing Ring :=
IsLocalization.isNoetherianRing ideal.primeCompl Ring
(inferInstance : IsNoetherianRing ℤ)
/-- The maximal ideal is exactly the principal ideal generated by two. -/
theorem maximalIdeal_eq_span :
IsLocalRing.maximalIdeal Ring = Ideal.span {(2 : Ring)} := by
calc
IsLocalRing.maximalIdeal Ring = Ideal.map (algebraMap ℤ Ring) ideal :=
(Localization.AtPrime.map_eq_maximalIdeal (I := ideal)).symm
_ = Ideal.span {(2 : Ring)} := by
simp only [ideal, Ideal.map_span, Set.image_singleton, map_ofNat]
theorem two_ne_zero : (2 : Ring) ≠ 0 := by norm_num
theorem two_not_isUnit : ¬ IsUnit (2 : Ring) := by
rw [← mem_nonunits_iff, ← IsLocalRing.mem_maximalIdeal, maximalIdeal_eq_span]
exact Ideal.mem_span_singleton_self _
/-- The concrete localization is a discrete valuation ring with uniformizer
(two); this conclusion is obtained from its proved principal maximal ideal. -/
instance isDiscreteValuationRing : IsDiscreteValuationRing Ring :=
LocalDVR.isDiscreteValuationRing_of_maximalIdeal_eq_span two_ne_zero
maximalIdeal_eq_span
@[simp] theorem mem_ideal_iff_even (z : ℤ) : z ∈ ideal ↔ Even z := by
simp only [ideal, Ideal.mem_span_singleton, even_iff_two_dvd]
/-- Exactly the odd integers become units. -/
theorem isUnit_algebraMap_iff_odd (z : ℤ) :
IsUnit (algebraMap ℤ Ring z) ↔ Odd z := by
rw [IsLocalization.AtPrime.isUnit_to_map_iff Ring ideal z]
change z ∉ ideal ↔ Odd z
rw [mem_ideal_iff_even, Int.not_even_iff_odd]
theorem isUnit_intCast_of_odd (z : ℤ) (hz : Odd z) : IsUnit (z : Ring) := by
simpa using (isUnit_algebraMap_iff_odd z).mpr hz
theorem isUnit_natCast_of_odd (u : ℕ) (hu : Odd u) : IsUnit (u : Ring) := by
simpa only [Int.cast_natCast] using
isUnit_intCast_of_odd (u : ℤ) ((Int.odd_coe_nat u).mpr hu)
end CirculantHadamard.IntegerTwoLocal
end
end OAI
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