Theorems · Theorem · algebraic geometry

AlgebraicGeometry.exists_mem_of_isClosed_of_nonempty

∀ {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.IsCofilteredOrEmpty I]
  [∀ {i j : I} (f : i ⟶ j), AlgebraicGeometry.IsAffineHom (D.map f)] (Z : (i : I) → Set ↥(D.obj i)),
  (∀ (i : I), IsClosed (Z i)) →
    (∀ (i : I), (Z i).Nonempty) →
      (∀ (i : I), IsCompact (Z i)) →
        (∀ {i i' : I} (f : i ⟶ i'), Set.MapsTo (⇑(D.map f)) (Z i) (Z i')) → ∃ s, ∀ (i : I), (c.π.app i) s ∈ Z i

Suppose we have a cofiltered diagram of schemes whose transition maps are affine. The limit of a family of compatible nonempty quasicompact closed sets in the diagram is also nonempty.

Defined in

Mathlib.AlgebraicGeometry.AffineTransitionLimit

Cited by

1 results in Mathlib

Foundations

Depth 235 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.IsCofilteredOrEmptyAlgebraicGeometry.IsAffineHom