Theorems · Theorem · algebraic geometry

AlgebraicGeometry.isSeparated_iff

∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y),
  AlgebraicGeometry.IsSeparated f ↔
    autoParam (AlgebraicGeometry.IsClosedImmersion (CategoryTheory.Limits.pullback.diagonal f))
      AlgebraicGeometry.IsSeparated.isClosedImmersion_diagonal._autoParam

Defined in: Mathlib.AlgebraicGeometry.Morphisms.Separated
Cited by: 2 results in Mathlib
Foundations: Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites7

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
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
CategoryTheory.Limits.pullback.diagonalObjstatement · cited by 80
CategoryTheory.Limits.pullback.diagonalstatement and proof · cited by 67
AlgebraicGeometry.IsClosedImmersionstatement and proof · cited by 49
AlgebraicGeometry.IsSeparatedstatement and proof · cited by 31
AlgebraicGeometry.IsSeparated.casesOnproof · cited by 1

Cited by2

Results whose statement or proof uses this declaration.

AlgebraicGeometry.IsSeparated.isSeparated_eq_diagonal_isClosedImmersionproof · cited by 0
AlgebraicGeometry.Scheme.isSeparated_iff_isClosedImmersion_prod_liftproof · cited by 0