openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Projective /ProjectiveMapEvaluation.lean
| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Projective.SectionMaps | |
| namespace OAI | |
| section | |
| noncomputable section | |
| section | |
| namespace MaximalSeshadri.Projective | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory HomogeneousLocalization HomogeneousIdeal | |
| universe u | |
| variable {A B R σ τ : Type u} [CommRing A] [CommRing B] [CommRing R] | |
| [SetLike σ A] [AddSubgroupClass σ A] [SetLike τ B] [AddSubgroupClass τ B] | |
| {𝒜 : ℕ → σ} [GradedRing 𝒜] {ℬ : ℕ → τ} [GradedRing ℬ] | |
| lemma evalAway_map (F : 𝒜 →+*ᵍ ℬ) (f : B →+* R) {s : A} {d : ℕ} | |
| (hs : s ∈ 𝒜 d) (hu : IsUnit (f (F s))) : | |
| (evalAway f (F s) hu).comp (Away.map F s) = | |
| evalAway (f.comp F.toRingHom) s hu := by | |
| apply RingHom.ext | |
| intro x | |
| obtain ⟨n, a, ha, rfl⟩ := Away.mk_surjective 𝒜 hs x | |
| apply (hu.pow n).mul_right_cancel | |
| simp only [RingHom.comp_apply, Away.map_mk] | |
| rw [evalAway_mk_clear] | |
| exact (evalAway_mk_clear (f.comp F.toRingHom) hs hu n a ha).symm | |
| lemma fromUnitCoordinate_comp_Proj_map (F : 𝒜 →+*ᵍ ℬ) | |
| (hF : ℬ₊ ≤ 𝒜₊.map F) (f : B →+* R) {s : A} {d : ℕ} | |
| (hd : 0 < d) (hs : s ∈ 𝒜 d) (hu : IsUnit (f (F s))) : | |
| fromUnitCoordinate f hd (F.map_mem hs) hu ≫ Proj.map F hF = | |
| fromUnitCoordinate (f.comp F.toRingHom) hd hs hu := by | |
| rw [fromUnitCoordinate, Category.assoc] | |
| erw [Proj.awayι_comp_map F hF hd s hs, | |
| ← Category.assoc, ← Spec.map_comp] | |
| change Spec.map (CommRingCat.ofHom | |
| ((evalAway f (F s) hu).comp (Away.map F s))) ≫ _ = _ | |
| rw [evalAway_map F f hs] | |
| rfl | |
| lemma fromUnitCoordinate_comp_Proj_map_eq (F : 𝒜 →+*ᵍ ℬ) | |
| (hF : ℬ₊ ≤ 𝒜₊.map F) (f : B →+* R) {s : A} {d : ℕ} | |
| (hd : 0 < d) (hs : s ∈ 𝒜 d) (t : B) (ht : F s = t) | |
| (htmem : t ∈ ℬ d) (hu : IsUnit (f t)) (g : A →+* R) | |
| (hg : f.comp F.toRingHom = g) (hv : IsUnit (g s)) : | |
| fromUnitCoordinate f hd htmem hu ≫ Proj.map F hF = | |
| fromUnitCoordinate g hd hs hv := by | |
| subst t | |
| subst g | |
| exact fromUnitCoordinate_comp_Proj_map F hF f hd hs hu | |
| end | |
| end MaximalSeshadri.Projective | |
| end | |
| end | |
| end | |
| end OAI | |