instFiniteResidueFieldOfFormallyUnramified

English

If p ⊂ R and q ⊂ S lie over each other, then IsUnramifiedAt R q and the condition on the localized maps propagate along the tower.

Русский

Если p ⊂ R и q ⊂ S лежат друг над другом, то неразветвленность распространяется по касательной к локализации.

LaTeX

$$$\\text{IsUnramifiedAt } R p \\; \\land \\text{IsUnramifiedAt } A P \\Rightarrow \\text{IsUnramifiedAt } R P$$$

Lean4

@[stacks 00UW "(2)"]
instance [FormallyUnramified R S] : Module.Finite (ResidueField R) (ResidueField S) :=
  have : EssFiniteType R (ResidueField S) := .comp _ S _
  have : EssFiniteType (ResidueField R) (ResidueField S) := .of_comp R _ _
  FormallyUnramified.finite_of_free _ _

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