| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.GeneratedEtaleCoordinates |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.SheafExactness |
| import OAI.AlgebraicGeometry.Seshadri.Jets.PolynomialJets |
|
|
| namespace OAI |
|
|
|
|
| |
| section |
|
|
| namespace MaximalSeshadri.AlgebraicJets |
| noncomputable section |
| open MvPolynomial |
| variable {K S : Type} [Field K] [CommRing S] [Algebra K S] |
|
|
| lemma etale_aeval_reindex {ι κ : Type} (x : ι → S) (h : (aeval (R := K) x).toRingHom.Etale) |
| (e : κ ≃ ι) : (aeval (R := K) (fun i => x (e i))).toRingHom.Etale := by |
| have he := RingHom.Etale.respectsIso.2 (aeval (R := K) x).toRingHom |
| (renameEquiv K e).toRingEquiv h |
| convert he using 1 |
| apply RingHom.ext |
| intro p |
| change aeval (fun i => x (e i)) p = aeval x (rename e p) |
| exact (aeval_rename e x p).symm |
|
|
| lemma etale_aeval_center {ι : Type} (x : ι → S) |
| (h : (aeval (R := K) x).toRingHom.Etale) (ρ : S →ₐ[K] K) : |
| (aeval (R := K) (fun i => x i - algebraMap K S (ρ (x i)))).toRingHom.Etale := by |
| have he := RingHom.Etale.respectsIso.2 (aeval (R := K) x).toRingHom |
| (polynomialTranslate (fun i => ρ (x i))).symm.toRingEquiv h |
| convert he using 1 |
| apply ringHom_ext |
| · intro c; simp [polynomialTranslate] |
| · intro i; simp [polynomialTranslate] |
|
|
| theorem exists_centered_etale_coordinates (n : ℕ) |
| [Algebra.IsStandardSmoothOfRelativeDimension n K S] (ρ : S →ₐ[K] K) : |
| ∃ x : Fin n → S, (∀ i, ρ (x i) = 0) ∧ |
| (aeval (R := K) x).toRingHom.Etale := by |
| obtain ⟨κ, hκ, x, hcard, hx⟩ := exists_etale_coordinates (R := K) (S := S) n |
| let : Fintype κ := Fintype.ofFinite κ |
| let e : κ ≃ Fin n := Fintype.equivFinOfCardEq (by simpa using hcard) |
| let y := fun i : Fin n => x (e.symm i) |
| refine ⟨fun i => y i - algebraMap K S (ρ (y i)), ?_, |
| etale_aeval_center y (etale_aeval_reindex x hx e.symm) ρ⟩ |
| intro i |
| simp |
|
|
| end |
|
|
| noncomputable section |
| variable {R : Type*} [CommRing R] |
|
|
| lemma exists_away_map_le (I J p : Ideal R) [p.IsPrime] (hI : I.FG) |
| (hle : I.map (algebraMap R (Localization.AtPrime p)) ≤ |
| J.map (algebraMap R (Localization.AtPrime p))) : |
| ∃ s : R, s ∉ p ∧ I.map (algebraMap R (Localization.Away s)) ≤ |
| J.map (algebraMap R (Localization.Away s)) := by |
| classical |
| obtain ⟨t, ht⟩ := hI |
| have hden : ∀ x : t, ∃ s : p.primeCompl, s.val * x.val ∈ J := by |
| intro x |
| have hx : algebraMap R (Localization.AtPrime p) x.val ∈ |
| J.map (algebraMap R (Localization.AtPrime p)) := |
| hle (Ideal.mem_map_of_mem _ (ht ▸ Ideal.subset_span x.property)) |
| obtain ⟨s, hs, h⟩ := |
| (IsLocalization.algebraMap_mem_map_algebraMap_iff p.primeCompl _ J x.val).mp hx |
| exact ⟨⟨s, hs⟩, h⟩ |
| choose d hd using hden |
| let s : p.primeCompl := ∏ x : t, d x |
| refine ⟨s.val, s.property, Ideal.map_le_iff_le_comap.mpr ?_⟩ |
| rw [← ht, Ideal.span_le] |
| intro x hx |
| have hdiv : d ⟨x, hx⟩ ∣ s := Finset.dvd_prod_of_mem d (Finset.mem_univ _) |
| obtain ⟨c, hc⟩ := hdiv |
| have hmul : s.val * x ∈ J := by |
| have h := J.mul_mem_left c.val (hd ⟨x, hx⟩) |
| simpa only [hc, Submonoid.coe_mul, mul_assoc, mul_left_comm] using h |
| change algebraMap R (Localization.Away s.val) x ∈ |
| J.map (algebraMap R (Localization.Away s.val)) |
| apply (IsLocalization.algebraMap_mem_map_algebraMap_iff |
| (Submonoid.powers s.val) _ J x).mpr |
| exact ⟨s.val, Submonoid.mem_powers _, hmul⟩ |
|
|
| lemma exists_away_map_eq_of_le (I J p : Ideal R) [p.IsPrime] (hI : I.FG) |
| (hJI : J ≤ I) |
| (heq : I.map (algebraMap R (Localization.AtPrime p)) = |
| J.map (algebraMap R (Localization.AtPrime p))) : |
| ∃ s : R, s ∉ p ∧ I.map (algebraMap R (Localization.Away s)) = |
| J.map (algebraMap R (Localization.Away s)) := by |
| obtain ⟨s, hs, hle⟩ := exists_away_map_le I J p hI heq.le |
| exact ⟨s, hs, le_antisymm hle (Ideal.map_mono hJI)⟩ |
|
|
| end |
|
|
| noncomputable section |
| open IsLocalRing |
| variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S] |
|
|
| lemma unramified_map_under_local [Algebra.EssFiniteType R S] |
| [Algebra.FormallyUnramified R S] (q : Ideal S) [q.IsPrime] : |
| (q.under R).map (algebraMap R (Localization.AtPrime q)) = |
| q.map (algebraMap S (Localization.AtPrime q)) := by |
| let p := q.under R |
| let Rp := Localization.AtPrime p |
| let Sq := Localization.AtPrime q |
| let : Algebra Rp Sq := Localization.AtPrime.algebraOfLiesOver p q |
| let : Algebra.EssFiniteType Rp Sq := Algebra.EssFiniteType.of_comp R Rp Sq |
| let : Algebra.FormallyUnramified R Sq := Algebra.FormallyUnramified.comp R S Sq |
| let : Algebra.FormallyUnramified Rp Sq := |
| Algebra.FormallyUnramified.of_restrictScalars R Rp Sq |
| have hmax : (maximalIdeal Rp).map (algebraMap Rp Sq) = maximalIdeal Sq := |
| Algebra.FormallyUnramified.map_maximalIdeal |
| rw [← IsLocalization.AtPrime.map_eq_maximalIdeal p Rp, |
| Ideal.map_map, ← IsScalarTower.algebraMap_eq, |
| ← IsLocalization.AtPrime.map_eq_maximalIdeal q Sq] at hmax |
| exact hmax |
|
|
| theorem exists_unramified_fiber_neighborhood [IsNoetherianRing S] |
| [Algebra.EssFiniteType R S] [Algebra.FormallyUnramified R S] |
| (q : Ideal S) [q.IsPrime] : |
| ∃ s : S, s ∉ q ∧ q.map (algebraMap S (Localization.Away s)) = |
| ((q.under R).map (algebraMap R S)).map |
| (algebraMap S (Localization.Away s)) := by |
| apply exists_away_map_eq_of_le q ((q.under R).map (algebraMap R S)) q |
| (Ideal.fg_of_isNoetherianRing q) Ideal.map_comap_le |
| rw [Ideal.map_map, ← IsScalarTower.algebraMap_eq] |
| exact (unramified_map_under_local (R := R) q).symm |
|
|
| end |
|
|
| noncomputable section |
| open MvPolynomial |
| variable {K S : Type} [Field K] [CommRing S] [Algebra K S] |
|
|
| theorem exists_centered_etale_point_chart (n : ℕ) |
| [Algebra.IsStandardSmoothOfRelativeDimension n K S] (ρ : S →ₐ[K] K) : |
| ∃ (x : Fin n → S) (s : S), ρ s ≠ 0 ∧ (∀ i, ρ (x i) = 0) ∧ |
| ((algebraMap S (Localization.Away s)).comp (aeval (R := K) x).toRingHom).Etale ∧ |
| (RingHom.ker ρ).map (algebraMap S (Localization.Away s)) = |
| (idealOfVars (Fin n) K).map |
| ((algebraMap S (Localization.Away s)).comp (aeval (R := K) x).toRingHom) := by |
| obtain ⟨x, hx, he⟩ := exists_centered_etale_coordinates n ρ |
| let φ := aeval (R := K) x |
| let : Algebra (MvPolynomial (Fin n) K) S := φ.toRingHom.toAlgebra |
| let : IsScalarTower K (MvPolynomial (Fin n) K) S := |
| IsScalarTower.of_algebraMap_eq' φ.comp_algebraMap.symm |
| let : Algebra.Etale (MvPolynomial (Fin n) K) S := he |
| let : Algebra.IsStandardSmooth K S := |
| Algebra.IsStandardSmoothOfRelativeDimension.isStandardSmooth n (R := K) (S := S) |
| let : IsNoetherianRing S := Algebra.FiniteType.isNoetherianRing K S |
| let q := RingHom.ker ρ |
| let : q.IsMaximal := RingHom.ker_isMaximal_of_surjective ρ |
| (fun c => ⟨algebraMap K S c, by simp⟩) |
| have hp : q.under (MvPolynomial (Fin n) K) = idealOfVars (Fin n) K := by |
| rw [idealOfVars_eq_ker_constantCoeff] |
| have hc : ρ.toRingHom.comp φ.toRingHom = constantCoeff := by |
| apply ringHom_ext |
| · intro c; simp [φ] |
| · intro i; simpa [φ] using hx i |
| change RingHom.ker (ρ.toRingHom.comp φ.toRingHom) = _ |
| rw [hc] |
| obtain ⟨s, hs, hI⟩ := |
| exists_unramified_fiber_neighborhood (R := MvPolynomial (Fin n) K) q |
| refine ⟨x, s, hs, hx, ?_, ?_⟩ |
| · change (algebraMap (MvPolynomial (Fin n) K) (Localization.Away s)).Etale |
| exact RingHom.etale_algebraMap.mpr inferInstance |
| · rw [hp, Ideal.map_map] at hI |
| convert hI using 1 |
| |
| end |
| end MaximalSeshadri.AlgebraicJets |
| |
| namespace MaximalSeshadri.PointBlowup |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits |
| open MaximalSeshadri.AlgebraicJets |
| variable {K S : Type} [Field K] [CommRing S] [Algebra K S] |
| |
| end |
| end MaximalSeshadri.PointBlowup |
| |
| end |
| |
| |
| end OAI |
| |
