File size: 2,898 Bytes
a72760f | 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 66 67 68 69 | import OAI.NumberTheory.PiExponent.Approximation.ClosedPullbackUnit
import OAI.NumberTheory.PiExponent.Approximation.OpenPullbackUnit
import OAI.NumberTheory.PiExponent.LocalAlgebra.IdealModule
namespace OAI
namespace PiExponentSeshadri.ClosedPullbackUnitRestriction
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
variable {X Y : Scheme.{0}} (f : Y ⟶ X)
lemma restrict_unit_isIso_iff (U : X.Opens) (M : X.Modules) :
IsIso ((Scheme.Modules.restrictFunctor U.ι).map
((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M)) ↔
IsIso ((Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U)).unit.app
(M.restrict U.ι)) := by
let g := (Scheme.Modules.restrictFunctor U.ι).map
((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M)
let e := OpenBaseChange.iso f U ((Scheme.Modules.pullback f).obj M)
let k := (Scheme.Modules.pushforward (f ∣_ U)).map
((OpenBaseChange.leftSquare f U).hom.app M)
calc
IsIso g ↔ IsIso (g ≫ e.hom) := (isIso_comp_right_iff g e.hom).symm
_ ↔ IsIso (((Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U)).unit.app
(M.restrict U.ι)) ≫ k) := by rw [OpenBaseChange.unit_compatibility]
_ ↔ _ := isIso_comp_right_iff _ k
theorem unit_isIso_of_affine (M : X.Modules)
(h : ∀ U : X.affineOpens,
IsIso ((Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U.1)).unit.app
(M.restrict U.1.ι))) :
IsIso ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M) := by
apply IdealModule.isIso_of_affine_app
intro U
have hr := (restrict_unit_isIso_iff f U.1 M).mpr (h U)
have ht := Scheme.Modules.Hom.isIso_iff_isIso_app.mp hr ⊤
change IsIso (((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M).app
(U.1.ι ''ᵁ ⊤)) at ht
have he : U.1.ι ''ᵁ ⊤ = U.1 :=
(Scheme.Hom.image_top_eq_opensRange U.1.ι).trans (Scheme.Opens.opensRange_ι U.1)
rw [he] at ht
exact ht
lemma morphismRestrict_surjective (hf : Function.Surjective f) (U : X.Opens) :
Function.Surjective (f ∣_ U) := by
intro x
obtain ⟨y, hy⟩ := hf x.1
refine ⟨⟨y, ?_⟩, ?_⟩
· change f y ∈ U
simpa only [hy] using x.2
· apply Subtype.ext
exact (morphismRestrict_base_coe f U _).trans hy
theorem unit_isIso_of_affine_pushforward [IsClosedImmersion f]
(hf : Function.Surjective f) (M : X.Modules)
(h : ∀ U : X.affineOpens, ∃ N : (f ⁻¹ᵁ U.1).toScheme.Modules,
Nonempty (M.restrict U.1.ι ≅ (Scheme.Modules.pushforward (f ∣_ U.1)).obj N)) :
IsIso ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M) := by
apply unit_isIso_of_affine f M
intro U
obtain ⟨N, ⟨e⟩⟩ := h U
exact ClosedPullbackUnit.unit_isIso_of_pushforward (f ∣_ U.1)
(morphismRestrict_surjective f hf U.1) e
end
end PiExponentSeshadri.ClosedPullbackUnitRestriction
end OAI
|