Small étale site

Let $X$ be a scheme. The small étale site $X_{\acute et}$ is the site defined as follows:

• its objects are étale morphisms $U\to X$;
• a morphism from $V\to X$ to $U\to X$ is an $X$-morphism $V\to U$;
• a family $\{U_i\to U\}$ is covering when every map is étale and the family is jointly surjective on underlying points.

The word small means that only schemes étale over the fixed base $X$ are used as objects, rather than all schemes over $X$.

When $X=\operatorname{Spec}F$ for a field $F$, every finite separable extension $K/F$ supplies a covering object $\operatorname{Spec}K\to X$. This is the local setting in which a finite Galois extension becomes a torsor.