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aliases:

formally etale date: “2022-03-22 21:25” modification date: “Tuesday 22nd March 2022 21:25:56” title: formally etale

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formally etale

Idea: meant to look like a local isomorphism by inducing a pointwise isomorphism on tangent spaces.

Idea: \(X\to S\) is etale when for all \(T\to X\), a first-order thickening \(T_1 \to S\) lifts uniquely to \(T_1\to X\).

Link to Diagram

Equivalent characterizations:

The relative differentials \(\Omega_{X/S} = 0\) (the relative differentials vanish) and the conormal sheaf vanishes, \(\mathcal{C}_{X/S} = 0\).
\({{\Gamma}\qty{X; {\mathcal{O}}_X} }\to {{\Gamma}\qty{Y; {\mathcal{O}}_Y} }\) is a formally etale morphism of rings.

Trivially implies formally unramified. If \(f\) is additionally locally of finite presentation, then \(f\) is an etale morphism.