Theorems · Theorem · algebraic geometry

AlgebraicGeometry.exists_app_map_eq_map_of_isLimit

∀ {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} {U : (D.obj i).Opens},
  IsCompact ↑U →
    ∀ (s t : ↑((D.obj i).presheaf.obj (Opposite.op U))),
      (CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.app (c.π.app i) U)) s =
          (CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.app (c.π.app i) U)) t →
        ∃ j f,
          (CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.app (D.map f) U)) s =
            (CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.app (D.map f) U)) t

Defined in

Mathlib.AlgebraicGeometry.AffineTransitionLimit

Cited by

1 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.IsCofilteredAlgebraicGeometry.IsAffineHom