base_change — Mathlib

English

Localizing the base along with the target preserves étaleness: if R → S is étale, then R_m → S_m is étale after localization at M.

Русский

Локализация базы и цели сохраняет étale: если R → S étale, то после локализации R_m → S_m тоже étale.

LaTeX

$$$[\text{FormallyEtale}(R,S)] \Rightarrow \text{FormallyEtale}(R_m,S_m)$ where $R_m,S_m$ are localizations along $M$ and $M.map$.$$

Lean4

instance base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
  FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩

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