Let {R} be a ring. An {R}-algebra {S} is said to be étale if it is finitely presented and for every {R}-algebra {S'} and every nilpotent ideal {I \subset S'} (or ideal consisting of nilpotents), we have an isomorphism

\displaystyle \hom_R(S, S') \simeq \hom_R(S, S'/I).

In other words, given a homomorphism of {R}-algebras {S \rightarrow S'/I}, we can lift it uniquely to the “nilpotent thickening” {S'}.

The algebras étale over {R} form a category, {\mathrm{Et}(R)}; this is the étale site of {R}. For example, for a field, it consists of the category of all finite separable extensions. \’Etaleness is preserved under base-change, so for any morphism {R \rightarrow S}, there is a functor

\displaystyle \mathrm{Et}(R) \rightarrow \mathrm{Et}(S).

A basic property of étale morphisms is the following:

Theorem 1 Let {R} be a ring and {J \subset R} a square zero (or nilpotent) ideal. Then there is an equivalence of categories

\displaystyle \mathrm{Et}(R) \rightarrow \mathrm{Et}(R/J)

given by tensoring with {R/J}.

This result is often proved using the local structure theory for étale morphisms, but this is fairly difficult: as far as I know, the local structure theory requires Zariski’s Main Theorem for its proof. (Correction: as observed below, one only needs a portion of the local structure theory to make the argument, and that portion does not require ZMT.) Here is a more elementary argument. (more…)