instHasRingHomPropertyIsSmoothLocallyIsStandardSmooth

English

Domination plus equality after postcomposition implies equality of morphisms.

Русский

Доминирующее отображение и равенство после композиции означает равенство морфизмов.

LaTeX

$$ext_of_isDominant (f) (g) (ι) (hU)$$

Lean4

/-- The property of scheme morphisms `IsSmooth` is associated with the ring
homomorphism property `Locally IsStandardSmooth`. -/
instance : HasRingHomProperty @IsSmooth (Locally IsStandardSmooth) :=
  by
  apply HasRingHomProperty.locally_of_iff
  · exact isStandardSmooth_localizationPreserves.away
  · exact isStandardSmooth_stableUnderCompositionWithLocalizationAway
  · intro X Y f
    rw [isSmooth_iff]

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