Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.OpenCover.exists_of_isCofiltered_of_finite

∀ {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), QuasiSeparatedSpace ↥(D.obj i)] (𝒰 : c.pt.OpenCover) [∀ (i : 𝒰.I₀), AlgebraicGeometry.IsAffine (𝒰.X i)]
  [Finite 𝒰.I₀],
  ∃ i R f,
    ∃ (_ :
      CategoryTheory.Presieve.ofArrows (fun i => AlgebraicGeometry.Spec (R i)) f ∈
        AlgebraicGeometry.Scheme.zariskiPrecoverage.coverings (D.obj i)),
      ∃ g, ∀ (j : 𝒰.I₀), CategoryTheory.IsPullback (g j) (𝒰.f j) (f j) (c.π.app i)

Variant of Scheme.exists_isOpenCover_and_isAffine_of_finite in terms of Scheme.OpenCover.

Defined in

Mathlib.AlgebraicGeometry.AffineTransitionLimit

Cited by

0 results in Mathlib

Foundations

Depth 244 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.IsCofilteredAlgebraicGeometry.IsAffineHomCompactSpaceQuasiSeparatedSpaceAlgebraicGeometry.IsAffineFinite