| 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 |
|
|
