Theorems · Theorem · algebraic geometry

AlgebraicGeometry.isIso_pushoutSection_iff

∀ {X Y S T : AlgebraicGeometry.Scheme} {f : T ⟶ S} {g : Y ⟶ X} {iX : X ⟶ S} {iY : Y ⟶ T}
  (H : CategoryTheory.IsPullback g iY iX f) {US : S.Opens} {UT : T.Opens} {UX : X.Opens}
  (hUST : UT ≤ (TopologicalSpace.Opens.map f.base).obj US) (hUSX : UX ≤ (TopologicalSpace.Opens.map iX.base).obj US)
  {UY : Y.Opens} (hUY : UY = (TopologicalSpace.Opens.map g.base).obj UX ⊓ (TopologicalSpace.Opens.map iY.base).obj UT),
  CategoryTheory.IsIso (AlgebraicGeometry.pushoutSection H hUST hUSX hUY) ↔
    CategoryTheory.IsPushout (AlgebraicGeometry.Scheme.Hom.appLE iX US UX hUSX)
      (AlgebraicGeometry.Scheme.Hom.appLE f US UT hUST) (AlgebraicGeometry.Scheme.Hom.appLE g UX UY ⋯)
      (AlgebraicGeometry.Scheme.Hom.appLE iY UT UY ⋯)

Defined in

Mathlib.AlgebraicGeometry.Morphisms.Flat

Cited by

1 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound