Let be a ring. An -algebra is said to be *étale* if it is finitely presented and for every -algebra and every nilpotent ideal (or ideal consisting of nilpotents), we have an isomorphism

In other words, given a homomorphism of -algebras , we can lift it uniquely to the “nilpotent thickening” .

The algebras étale over form a category, ; this is the *étale site* of . 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 , there is a functor

A basic property of étale morphisms is the following:

Theorem 1Let be a ring and a square zero (or nilpotent) ideal. Then there is an equivalence of categories

given by tensoring with .

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…)