exists_isPrime_liesOver_of_faithfullyFlat

English

If B is faithfully flat over A, every prime ideal p of A lies under some prime P of B.

Русский

Если B плоско над A и Pprime —prime идеал над A, существует Pprime над B с P LiesOver p.

LaTeX

$$∃ P ∈ Spec(B) such that P.IsPrime ∧ P.LiesOver p$$

Lean4

/-- If `B` is faithfully flat over `A`, every prime of `A` comes from a prime of `B`. -/
theorem exists_isPrime_liesOver_of_faithfullyFlat (p : Ideal A) [p.IsPrime] :
    ∃ (P : Ideal B), P.IsPrime ∧ P.LiesOver p :=
  by
  obtain ⟨P, _, hP⟩ := (Ideal.comap_map_eq_self_iff_of_isPrime p).mp <| p.comap_map_eq_self_of_faithfullyFlat (B := B)
  exact ⟨P, inferInstance, ⟨hP.symm⟩⟩

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