affineAnd_containsIdentities — Mathlib

English

If P has HasAffineProperty with affineAnd(Q) and Q is stable under base change, then P is stable under base change.

Русский

Если P имеет свойство HasAffineProperty через affineAnd(Q) и Q устойчиво к базовым изменениям, тогда P устойчиво к базовому изменению.

LaTeX

$$$P\\text{ IsStableUnderBaseChange}$$$

Lean4

/-- If `Q` contains identities and respects isomorphisms (i.e. is satisfied by isomorphisms),
and `P` is affine locally defined by `affineAnd Q`, then `P` contains identities. -/
theorem affineAnd_containsIdentities {P : MorphismProperty Scheme.{u}} (hA : HasAffineProperty P (affineAnd Q))
    (hQi : RingHom.RespectsIso Q) (hQ : RingHom.ContainsIdentities Q) : P.ContainsIdentities where
  id_mem
    X := by
    rw [eq_targetAffineLocally P, targetAffineLocally_affineAnd_iff hQi]
    intro U hU
    exact ⟨hU, hQ _⟩

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