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  • formally etale

formally smooth

Idea: \(E \xrightarrow{f} B\) is formally smooth if tangent vectors \(\operatorname{Spec}k[{\varepsilon}]/{\varepsilon}^2 \to B\) to tangent vectors \(\operatorname{Spec}k[{\varepsilon}]/{\varepsilon}^2 \to E\).

Being formally unramified says there is at most one lift, and formally etale means an isomorphism of tangent spaces. Meant to look like a submersion.