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…)