File size: 1,662 Bytes
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 | import Mathlib
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry TopologicalSpace Order
lemma scheme_dimension_eq_point_dimension (X : Scheme) :
topologicalKrullDim X = Order.krullDim X :=
Order.krullDim_eq_of_orderIso (irreducibleSetEquivPoints (α := X))
lemma scheme_dimension_le_of_neighborhoods (X : Scheme) (d : WithBot ℕ∞)
(h : ∀ x : X, ∃ (U : Scheme) (f : U ⟶ X), IsOpenImmersion f ∧
x ∈ Set.range f ∧ topologicalKrullDim U ≤ d) : topologicalKrullDim X ≤ d := by
rw [scheme_dimension_eq_point_dimension, Order.krullDim_eq_iSup_coheight]
apply iSup_le
intro x
obtain ⟨U,f,hf,⟨y,rfl⟩,hd⟩ := h x
let := hf
rw [AlgebraicGeometry.coheight_eq_of_isOpenImmersion]
exact (Order.coheight_le_krullDim y).trans ((scheme_dimension_eq_point_dimension U) ▸ hd)
lemma subscheme_dimension_le_of_affine_neighborhoods {X : Scheme}
(I : X.IdealSheafData) (d : WithBot ℕ∞)
(h : ∀ x : I.subscheme, ∃ U : X.affineOpens, I.subschemeι x ∈ U.1 ∧
ringKrullDim (Γ(X,U.1) ⧸ I.ideal U) ≤ d) :
topologicalKrullDim I.subscheme ≤ d := by
apply scheme_dimension_le_of_neighborhoods
intro x
obtain ⟨U,hx,hd⟩ := h x
let := I.subschemeCover.map_prop U
refine ⟨_, I.subschemeCover.f U,inferInstance,?_,?_⟩
· change x ∈ (I.subschemeCover.f U).opensRange
rw [I.opensRange_subschemeCover_map]
exact hx
· exact (PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim _).le.trans hd
end
end MaximalSeshadri.Geometry
end
end
end
end OAI
|