Theorems · Definition · algebraic geometry

AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso

{X Y : AlgebraicGeometry.Scheme} →
  (f : X ⟶ Y) →
    [inst : AlgebraicGeometry.QuasiCompact f] →
      [inst_1 : AlgebraicGeometry.QuasiSeparated f] →
        {U V : AlgebraicGeometry.Scheme} →
          {iU : U ⟶ X} →
            {iV : V ⟶ X} →
              CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV) →
                [inst_2 : AlgebraicGeometry.QuasiCompact iU] →
                  [inst_3 : AlgebraicGeometry.QuasiSeparated iU] →
                    [inst_4 : AlgebraicGeometry.QuasiCompact iV] →
                      [inst_5 : AlgebraicGeometry.QuasiSeparated iV] →
                        AlgebraicGeometry.Scheme.Hom.normalization (CategoryTheory.CategoryStruct.comp iU f) ⨿
                            AlgebraicGeometry.Scheme.Hom.normalization (CategoryTheory.CategoryStruct.comp iV f) ≅
                          AlgebraicGeometry.Scheme.Hom.normalization f

The normalization of Y in a coproduct is isomorphic to the coproduct of the normalizations in each of the components.

Defined in

Mathlib.AlgebraicGeometry.Normalization

Cited by

15 results in Mathlib

Foundations

Depth 227 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AlgebraicGeometry.QuasiCompactAlgebraicGeometry.QuasiSeparatedAlgebraicGeometry.QuasiCompactAlgebraicGeometry.QuasiSeparatedAlgebraicGeometry.QuasiCompactAlgebraicGeometry.QuasiSeparated