Theorems · Theorem · algebraic geometry

AlgebraicGeometry.exists_map_eq_top

∀ {I : Type u} [inst : CategoryTheory.Category.{u, u} I] (D : CategoryTheory.Functor I AlgebraicGeometry.Scheme)
  (c : CategoryTheory.Limits.Cone D) (hc : CategoryTheory.Limits.IsLimit c) [CategoryTheory.IsCofiltered I]
  [∀ {i j : I} (f : i ⟶ j), AlgebraicGeometry.IsAffineHom (D.map f)] [∀ (i : I), CompactSpace ↥(D.obj i)] {i : I}
  (U : (D.obj i).Opens),
  (TopologicalSpace.Opens.map (c.π.app i).base).obj U = ⊤ →
    ∃ j fji, (TopologicalSpace.Opens.map (D.map fji).base).obj U = ⊤

Let { Dᵢ } be a cofiltered diagram of compact schemes with affine transition maps. If U ⊆ Dⱼ contains the image of limᵢ Dᵢ ⟶ Dⱼ, then it contains the image of some Dₖ ⟶ Dⱼ.

Defined in

Mathlib.AlgebraicGeometry.AffineTransitionLimit

Cited by

5 results in Mathlib

Foundations

Depth 237 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.IsCofilteredAlgebraicGeometry.IsAffineHomCompactSpace