instQuasiSoberCarrierCarrierCommRingCat

English

If f: X → Y is an open immersion and Y is reduced, then X is reduced.

Русский

Если f: X → Y является открытым вложением и Y редуцирован, то X редуцирован.

LaTeX

$$$\text{IsReduced} X \text{ given } f \text{ open immersion and } \text{IsReduced } Y$$$

Lean4

instance : QuasiSober X :=
  by
  apply (config := { allowSynthFailures := true })
    quasiSober_of_open_cover (Set.range fun x => Set.range <| (X.affineCover.f x).base)
  · rintro ⟨_, i, rfl⟩; exact (X.affineCover.map_prop i).base_open.isOpen_range
  · rintro ⟨_, i, rfl⟩
    exact
      @IsOpenEmbedding.quasiSober _ _ _ _ _
        (X.affineCover.map_prop i).base_open.isEmbedding.toHomeomorph.symm.isOpenEmbedding PrimeSpectrum.quasiSober
  · rw [Set.top_eq_univ, Set.sUnion_range, Set.eq_univ_iff_forall]
    intro x; exact ⟨_, ⟨_, rfl⟩, X.affineCover.covers x⟩

Похожие утверждения