English
If every component f i is formally unramified over R, then the Pi-type (i → f i) is formally unramified over R.
Русский
Если каждая компонента f i формально неразветвлена над R, то Π i, f i формально неразвлена над R.
LaTeX
$$$[\\forall i, \\mathrm{FormallyUnramified}\\; R\\; (f i)] \\Rightarrow \\mathrm{FormallyUnramified}\\; R\\; ((i:\\,I) \\to f i).$$$
Lean4
/-- Given an algebra between two field extensions `L` and `E` of a field `K` with a valuation `v`,
create an algebra between their two rings of integers. -/
instance algebra : Algebra (integralClosure v.valuationSubring L) (integralClosure v.valuationSubring E) :=
RingHom.toAlgebra
{ toFun := fun k => ⟨algebraMap L E k, IsIntegral.algebraMap k.2⟩
map_zero' := Subtype.ext <| by simp only [Subalgebra.coe_zero, map_zero]
map_one' := Subtype.ext <| by simp only [Subalgebra.coe_one, map_one]
map_add' := fun x y => Subtype.ext <| by simp only [map_add, Subalgebra.coe_add]
map_mul' := fun x y => Subtype.ext <| by simp only [Subalgebra.coe_mul, map_mul] }
