Theorems · Theorem · algebraic geometry

AlgebraicGeometry.formallyUnramified_iff

∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y),
  AlgebraicGeometry.FormallyUnramified f ↔
    ∀ {U : Y.Opens},
      AlgebraicGeometry.IsAffineOpen U →
        ∀ {V : X.Opens},
          AlgebraicGeometry.IsAffineOpen V →
            ∀ (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U),
              (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.appLE f U V e)).FormallyUnramified

Defined in: Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified
Cited by: 1 results in Mathlib
Foundations: Depth 141 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites24

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.Functor.objstatement and proof · cited by 19,642
Oppositestatement · cited by 8,081
TopCat.carrierstatement · cited by 3,184
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
CommRingCatstatement · cited by 2,333
TopologicalSpace.Opensstatement · cited by 2,040
AlgebraicGeometry.PresheafedSpace.carrierstatement · cited by 2,020
AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement · cited by 1,988
AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement · cited by 1,892
AlgebraicGeometry.Scheme.toLocallyRingedSpacestatement · cited by 1,734
AlgebraicGeometry.Scheme.Opensstatement and proof · cited by 1,149

Cited by1

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Etale.iff_flat_and_formallyUnramifiedproof · cited by 1