English
The glued-to-sheaf presheaf is locally injective.
Русский
Склеенная кшейф прешепф бесклейно инъективна локально.
LaTeX
$$Sheaf.IsLocallyInjective (yonedaGluedToSheaf hf)$$
Lean4
instance : Sheaf.IsLocallyInjective (yonedaGluedToSheaf hf) where
equalizerSieve_mem := by
rintro ⟨U⟩ (α β : U ⟶ _) h
replace h : (yonedaGluedToSheaf hf).val.app _ α = (yonedaGluedToSheaf hf).val.app _ β := h
have mem := (glueData hf).openCover.mem_grothendieckTopology
refine
GrothendieckTopology.superset_covering _ ?_
(zariskiTopology.intersection_covering (zariskiTopology.pullback_stable α mem)
(zariskiTopology.pullback_stable β mem))
rintro V (γ : _ ⟶ U) ⟨⟨W₁, a, _, ⟨i⟩, fac₁⟩, ⟨W₂, b, _, ⟨j⟩, fac₂⟩⟩
change γ ≫ α = γ ≫ β
replace h : (yonedaGluedToSheaf hf).val.app _ (γ ≫ α) = (yonedaGluedToSheaf hf).val.app _ (γ ≫ β) := by simp [h]
rw [← fac₁, ← fac₂] at h ⊢
apply comp_toGlued_eq
simpa [Scheme.GlueData.openCover_X, yonedaEquiv_naturality] using h
