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12dee50 | 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 | 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
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