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b797a5b | 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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.NumberTheory.PiExponent.Polynomials.GradedLocalizationExact
namespace OAI
namespace PiExponent.GradedLocalizationFree
noncomputable section
open scoped BigOperators
open GradedLocalizationExact
variable {R J σR : Type*} [CommRing R] [Fintype J]
[SetLike σR R] [AddSubgroupClass σR R]
def coordinateMap (a : R) :
LocalizedModule (Submonoid.powers a) (J → R) →ₗ[R]
(J → LocalizedModule (Submonoid.powers a) R) :=
LinearMap.pi (fun j => localizedMap a (LinearMap.proj j))
omit [Fintype J] in
@[simp] theorem coordinateMap_fraction (a : R) (v : J → R) (n : ℕ) (j : J) :
coordinateMap a (fraction a v n) j = fraction a (v j) n := by
change localizedMap a (LinearMap.proj (φ := fun _ : J => R) j) (fraction a v n) = _
exact localizedMap_fraction a (LinearMap.proj (φ := fun _ : J => R) j) v n
theorem coordinateMap_injective (a : R) : Function.Injective (coordinateMap (J := J) a) := by
classical
apply LinearMap.ker_eq_bot.mp
apply le_antisymm ?_ bot_le
intro z hz
change coordinateMap a z = 0 at hz
change z = 0
induction z using LocalizedModule.induction_on with
| _ v s =>
obtain ⟨n, hn⟩ := s.property
have hs : s = powerDenominator a n := Subtype.ext hn.symm
subst s
change coordinateMap a (fraction a v n) = 0 at hz
have hcoord : ∀ j, ∃ k : ℕ, a ^ k • v j = 0 := by
intro j
apply (fraction_eq_zero a (v j) n).mp
have h := congrFun hz j
simpa only [coordinateMap_fraction, Pi.zero_apply] using h
choose k hk using hcoord
let K : ℕ := ∑ j, k j
apply (fraction_eq_zero a v n).mpr
refine ⟨K, ?_⟩
funext j
have hj : k j ≤ K := Finset.single_le_sum (fun i _ => Nat.zero_le (k i))
(Finset.mem_univ j)
change a ^ K • v j = 0
calc
a ^ K • v j = (a ^ (K - k j) * a ^ k j) • v j := by
rw [← pow_add, Nat.sub_add_cancel hj]
_ = 0 := by rw [mul_smul, hk j, smul_zero]
variable (𝒜 : ℤ → σR) [SetLike.GradedMonoid 𝒜]
(w : J → ℤ) (a : R) (e : ℤ) (ha : a ∈ 𝒜 e) (d : ℤ)
def coordinatePieces :
degreePiece 𝒜 (ShiftedFreeGrading.piece 𝒜 w) a e ha d →+
(∀ j, degreePiece 𝒜 𝒜 a e ha (d - w j)) where
toFun z j := ⟨coordinateMap a z.val j, by
obtain ⟨n, v, hv, hz⟩ := z.property
refine ⟨n, v j, ?_, ?_⟩
· have h := hv j
change v j ∈ 𝒜 (d + (n : ℤ) * e - w j) at h
convert h using 1
ring_nf
· rw [hz, coordinateMap_fraction]⟩
map_zero' := by
funext j
exact Subtype.ext (congrFun (map_zero (coordinateMap a)) j)
map_add' z z' := by
funext j
exact Subtype.ext (congrFun (map_add (coordinateMap a) z.val z'.val) j)
@[simp] theorem coordinatePieces_coe
(z : degreePiece 𝒜 (ShiftedFreeGrading.piece 𝒜 w) a e ha d) (j : J) :
(coordinatePieces 𝒜 w a e ha d z j).val = coordinateMap a z.val j := rfl
theorem coordinatePieces_injective : Function.Injective (coordinatePieces 𝒜 w a e ha d) := by
intro z z' h
apply Subtype.ext
apply coordinateMap_injective a
funext j
exact congrArg Subtype.val (congrFun h j)
theorem coordinatePieces_surjective : Function.Surjective (coordinatePieces 𝒜 w a e ha d) := by
classical
intro z
have hz : ∀ j, ∃ (n : ℕ) (m : R),
m ∈ 𝒜 (d - w j + (n : ℤ) * e) ∧ (z j).val = fraction a m n :=
fun j => (z j).property
choose n m hm hzm using hz
let N : ℕ := ∑ j, n j
have hn (j : J) : n j ≤ N := Finset.single_le_sum
(fun i _ => Nat.zero_le (n i)) (Finset.mem_univ j)
let v : J → R := fun j => a ^ (N - n j) • m j
have hv : v ∈ ShiftedFreeGrading.piece 𝒜 w (d + (N : ℤ) * e) := by
intro j
have h := power_smul_mem 𝒜 𝒜 a e ha (hm j) (N - n j)
change v j ∈ 𝒜 (d + (N : ℤ) * e - w j)
dsimp only [v]
convert h using 1
rw [Nat.cast_sub (hn j)]
ring_nf
refine ⟨⟨fraction a v N, N, v, hv, rfl⟩, ?_⟩
funext j
apply Subtype.ext
change coordinateMap a (fraction a v N) j = (z j).val
rw [coordinateMap_fraction, hzm]
change fraction a (a ^ (N - n j) • m j) N = fraction a (m j) (n j)
have hc := fraction_cancel a (m j) (n j) (N - n j)
simpa only [Nat.add_sub_of_le (hn j)] using hc
def shiftedFreeEquiv :
degreePiece 𝒜 (ShiftedFreeGrading.piece 𝒜 w) a e ha d ≃+
(∀ j, degreePiece 𝒜 𝒜 a e ha (d - w j)) :=
AddEquiv.ofBijective (coordinatePieces 𝒜 w a e ha d)
⟨coordinatePieces_injective 𝒜 w a e ha d,
coordinatePieces_surjective 𝒜 w a e ha d⟩
@[simp] theorem shiftedFreeEquiv_apply
(z : degreePiece 𝒜 (ShiftedFreeGrading.piece 𝒜 w) a e ha d) (j : J) :
(shiftedFreeEquiv 𝒜 w a e ha d z j).val = coordinateMap a z.val j := rfl
end
end PiExponent.GradedLocalizationFree
end OAI
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