| import OAI.AlgebraicGeometry.StableCoordinate.LocalizationLND |
| import OAI.AlgebraicGeometry.StableCoordinate.DerivationScalarUpgrade |
| import OAI.AlgebraicGeometry.StableCoordinate.UnivariateLND |
| import Mathlib.RingTheory.Localization.Away.Basic |
|
|
| namespace OAI |
|
|
| /-! |
| # Constant coefficients forced by a locally nilpotent derivation on a chart |
|
|
| This is the complete one-chart part of the §5 rigidity argument. Starting |
| with a locally nilpotent derivation of `A`, not with an assumed derivation on |
| the chart, we localize at an invariant element, transport through a polynomial |
| chart, and enlarge the coefficient ring to its localization. The one-variable |
| LND theorem then forces a vertical coefficient to be constant on the chart. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace StableCoordinate.ChartLNDConstant |
|
|
| open Polynomial LocalizationLND |
|
|
| variable {k R A L S : Type*} |
| [CommRing k] [CommRing R] [CommRing A] [CommRing L] [CommRing S] |
| [Algebra k R] [Algebra k A] [Algebra k L] [Algebra k S] |
| [Algebra A L] [IsScalarTower k A L] |
| [Algebra R S] |
| [IsDomain S] [CharZero S] |
| (i : R →ₐ[k] A) (a : R) |
| [IsLocalization.Away (i a) L] [IsLocalization.Away a S] |
|
|
| / |
| of the polynomial chart variable, multiplied by the inverted coefficient. |
| All localization, transport, scalar-linearity, and nilpotence obligations are |
| proved from the original derivation and the displayed chart identities. -/ |
| theorem chart_coefficient_eq_C |
| (D : Derivation k A A) |
| (hD : ∀ x : A, ∃ n : ℕ, ((D : A → A)^[n]) x = 0) |
| (hi : ∀ r : R, D (i r) = 0) |
| (e : L ≃ₐ[k] S[X]) |
| (he : ∀ r : R, e (algebraMap A L (i r)) = C (algebraMap R S r)) |
| (b q : A) |
| (hb : e (algebraMap A L b) = X) |
| (hDb : D b = i a * q) : |
| e (algebraMap A L q) = C ((e (algebraMap A L q)).coeff 0) := by |
| have hden : ∀ m : Submonoid.powers (i a), D (m : A) = 0 := by |
| rintro ⟨m, hm⟩ |
| obtain ⟨n, rfl⟩ := (Submonoid.mem_powers_iff m (i a)).mp hm |
| simp only [Derivation.leibniz_pow, hi a, smul_zero] |
| let DL : Derivation k L L := |
| localizeTo D (Submonoid.powers (i a)) hden L |
| let E : Derivation k S[X] S[X] := transport DL e |
| have hEmap (x : A) : |
| E (e (algebraMap A L x)) = e (algebraMap A L (D x)) := by |
| dsimp only [E, DL] |
| rw [transport_apply, localizeTo_algebraMap] |
| have hELND : ∀ p : S[X], ∃ n : ℕ, ((E : S[X] → S[X])^[n]) p = 0 := |
| transport_locallyNilpotent DL e |
| (localizeTo_locallyNilpotent D (Submonoid.powers (i a)) hden L hD) |
| have hEC : ∀ s : S, E (C s) = 0 := by |
| apply DerivationScalarUpgrade.annihilates_localization |
| (Submonoid.powers a) E (C : S →+* S[X]) |
| intro r |
| rw [← he r, hEmap, hi, map_zero, map_zero] |
| let ES : Derivation S S[X] S[X] := DerivationScalarUpgrade.upgrade E hEC |
| have hESLND : ∀ p : S[X], ∃ n : ℕ, ((ES : S[X] → S[X])^[n]) p = 0 := |
| DerivationScalarUpgrade.upgrade_locallyNilpotent E hEC hELND |
| have hESX : ES X = C (algebraMap R S a) * e (algebraMap A L q) := by |
| change E X = _ |
| rw [← hb, hEmap, hDb, map_mul, map_mul, he] |
| have hconst := UnivariateLND.apply_X_eq_C_of_locallyNilpotent ES hESLND |
| rw [hESX, coeff_C_mul, C_mul] at hconst |
| exact ((IsLocalization.Away.algebraMap_isUnit (S := S) a).map |
| (C : S →+* S[X])).mul_left_cancel hconst |
|
|
| / |
| theorem exists_chart_coefficient |
| (D : Derivation k A A) |
| (hD : ∀ x : A, ∃ n : ℕ, ((D : A → A)^[n]) x = 0) |
| (hi : ∀ r : R, D (i r) = 0) |
| (e : L ≃ₐ[k] S[X]) |
| (he : ∀ r : R, e (algebraMap A L (i r)) = C (algebraMap R S r)) |
| (b q : A) |
| (hb : e (algebraMap A L b) = X) |
| (hDb : D b = i a * q) : |
| ∃ s : S, e (algebraMap A L q) = C s := |
| ⟨_, chart_coefficient_eq_C i a D hD hi e he b q hb hDb⟩ |
|
|
| end StableCoordinate.ChartLNDConstant |
|
|
| end |
|
|
| end OAI |
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|
