| 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 | |