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Theorems · Definition · algebraic geometry

AlgebraicGeometry.Scheme.Hom.normalizationPullback

{X S Y : AlgebraicGeometry.Scheme} →
  (f : X ⟶ S) →
    (g : Y ⟶ S) →
      [inst : AlgebraicGeometry.QuasiCompact f] →
        [inst_1 : AlgebraicGeometry.QuasiSeparated f] →
          AlgebraicGeometry.Scheme.Hom.normalization (CategoryTheory.Limits.pullback.snd f g) ⟶
            CategoryTheory.Limits.pullback (AlgebraicGeometry.Scheme.Hom.fromNormalization f) g

The comparison lemma between the normalization of the pullback to the pullback of the normalization. This is an isomorphism when g is smooth.

Defined in
Mathlib.AlgebraicGeometry.Normalization
Cited by
5 results in Mathlib
Foundations
Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
AlgebraicGeometry.QuasiCompactAlgebraicGeometry.QuasiSeparated
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

AlgebraicGeometry.Scheme.Hom.toNormalization_normalizationPullback_fst · cited by 2Hom.toNormalization_norma…AlgebraicGeometry.Scheme.Hom.normalizationPullback_snd · cited by 2Hom.normalizationPullback…AlgebraicGeometry.Scheme.Hom.exists_mem_and_isIso_morphismRestrict_toNormalization · cited by 1Hom.exists_mem_and_isIso_…AlgebraicGeometry.Scheme.Hom.normalizationPullback_snd_assoc · cited by 0Hom.normalizationPullback…AlgebraicGeometry.Scheme.Hom.toNormalization_normalizationPullback_fst_assoc · cited by 0Hom.toNormalization_norma…Quiver.Hom · cited by 32603Quiver.HomCategoryTheory.CategoryStruct.id · cited by 6235CategoryStruct.idAlgebraicGeometry.Scheme · cited by 2540AlgebraicGeometry.SchemeCategoryTheory.Limits.pullback · cited by 864Limits.pullbackCategoryTheory.Limits.pullback.snd · cited by 637pullback.sndCategoryTheory.Limits.pullback.map · cited by 156pullback.mapAlgebraicGeometry.QuasiCompact · cited by 102AlgebraicGeometry.QuasiCo…AlgebraicGeometry.QuasiSeparated · cited by 52AlgebraicGeometry.QuasiSe…AlgebraicGeometry.Scheme.Hom.normalization · cited by 41Hom.normalizationAlgebraicGeometry.Scheme.Hom.toNormalization · cited by 28Hom.toNormalizationAlgebraicGeometry.Scheme.Hom.fromNormalization · cited by 28Hom.fromNormalizationAlgebraicGeometry.Scheme.Hom.normalizationDesc · cited by 10Hom.normalizationDescHom.normalizationPullbackCITED BYCITES

Cites12

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Cited by5

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