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 _⟩