Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.exists_hom_comp_eq_comp_of_locallyOfFiniteType

∀ {I : Type u} [inst : CategoryTheory.Category.{u, u} I] {S X : AlgebraicGeometry.Scheme}
  (D : CategoryTheory.Functor I AlgebraicGeometry.Scheme) (t : D ⟶ (CategoryTheory.Functor.const I).obj S) (f : X ⟶ S)
  (c : CategoryTheory.Limits.Cone D) (hc : CategoryTheory.Limits.IsLimit c) [∀ (i : I), CompactSpace ↥(D.obj i)]
  [AlgebraicGeometry.LocallyOfFiniteType f] [CategoryTheory.IsCofiltered I]
  [∀ {i j : I} (f : i ⟶ j), AlgebraicGeometry.IsAffineHom (D.map f)] {i : I} (a b : D.obj i ⟶ X),
  t.app i = CategoryTheory.CategoryStruct.comp a f →
    t.app i = CategoryTheory.CategoryStruct.comp b f →
      CategoryTheory.CategoryStruct.comp (c.π.app i) a = CategoryTheory.CategoryStruct.comp (c.π.app i) b →
        ∃ k hik, CategoryTheory.CategoryStruct.comp (D.map hik) a = CategoryTheory.CategoryStruct.comp (D.map hik) b

Defined in

Mathlib.AlgebraicGeometry.AffineTransitionLimit

Cited by

2 results in Mathlib

Foundations

Depth 246 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCompactSpaceAlgebraicGeometry.LocallyOfFiniteTypeCategoryTheory.IsCofilteredAlgebraicGeometry.IsAffineHom