Theorems · Theorem · algebraic geometry

AlgebraicGeometry.ext_of_isDominant

∀ {W X Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsReduced X] {f g : X ⟶ Y} [Y.IsSeparated] (ι : W ⟶ X)
  [AlgebraicGeometry.IsDominant ι],
  CategoryTheory.CategoryStruct.comp ι f = CategoryTheory.CategoryStruct.comp ι g → f = g

Suppose f g : X ⟶ Y where X is a reduced scheme and Y is a separated scheme. Then f = g if ι ≫ f = ι ≫ g for some dominant ι. Also see ext_of_isDominant_of_isSeparated for the general version over arbitrary bases.

Defined in: Mathlib.AlgebraicGeometry.Morphisms.Separated
Cited by: 1 results in Mathlib
Foundations: Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.IsReduced AlgebraicGeometry.Scheme.IsSeparated AlgebraicGeometry.IsDominant

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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.CategoryStruct.compstatement and proof · cited by 17,999
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
CategoryTheory.Limits.terminal.fromproof · cited by 77
AlgebraicGeometry.IsDominantstatement and proof · cited by 43
AlgebraicGeometry.IsReducedstatement and proof · cited by 38
AlgebraicGeometry.Scheme.IsSeparatedstatement and proof · cited by 7
CategoryTheory.Limits.terminal.hom_extproof · cited by 4
AlgebraicGeometry.ext_of_isDominant_of_isSeparatedproof · cited by 3

Cited by1

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Scheme.PartialMap.isOver_toRationalMap_iff_of_isSeparatedproof · cited by 0