Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_fst

∀ {X Y Z : AlgebraicGeometry.Scheme} (𝒰 : X.OpenCover) (f : X ⟶ Z) (g : Y ⟶ Z)
  [inst : ∀ (i : 𝒰.I₀), CategoryTheory.Limits.HasPullback (CategoryTheory.CategoryStruct.comp (𝒰.f i) f) g] (i : 𝒰.I₀),
  CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso 𝒰 f g i).hom
      (CategoryTheory.Limits.pullback.fst (CategoryTheory.CategoryStruct.comp (𝒰.f i) f) g) =
    CategoryTheory.Limits.pullback.snd (AlgebraicGeometry.Scheme.Pullback.p1 𝒰 f g) (𝒰.f i)

Defined in: Mathlib.AlgebraicGeometry.Pullbacks
Cited by: 1 results in Mathlib
Foundations: Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Limits.HasPullback

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Cites21

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
CategoryTheory.Iso.homstatement · cited by 7,684
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
CategoryTheory.Limits.pullbackstatement · cited by 864
CategoryTheory.PreZeroHypercover.I₀statement and proof · cited by 763
CategoryTheory.PreZeroHypercover.Xstatement · cited by 649
CategoryTheory.Limits.pullback.fststatement and proof · cited by 639
CategoryTheory.Limits.pullback.sndstatement and proof · cited by 637
CategoryTheory.PreZeroHypercover.fstatement and proof · cited by 542
AlgebraicGeometry.IsOpenImmersionstatement · cited by 476
CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercoverstatement and proof · cited by 469

Cited by1

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AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_fst_assocproof · cited by 0