Jointly surjective étale families form a Grothendieck pretopology

By [F1], arrows in the small étale category are all morphisms of $X$-schemes. Identity maps and composites of $X$-scheme morphisms are again $X$-scheme morphisms, so the stated objects and arrows form a category without requiring every arrow to be étale. The identity $U\to U$ is étale by the definition of étale morphism and has image all of $U$, so its identity family is a declared cover.

Let $U_i\to U$ be a declared cover and let $V\to U$ be any arrow. Each pullback $U_i{\times }_{U}V$ exists by [F3], and its projection to $V$ is étale by base-change stability in [F2]. Fix a point $v\in V$ and write $u$ for its image. Joint surjectivity in [F1] gives an index $i$ and a point $u_i\in U_i$ above $u$. The residue maps give field extensions $\kappa (u)\to \kappa (u_i)$ and $\kappa (u)\to \kappa (v)$, so $\kappa (u_i){\otimes }_{\kappa (u)}\kappa (v)$ is a nonzero ring: it is the tensor product of two nonzero vector spaces over the field $\kappa (u)$. Under [A1], this nonzero ring has a prime ideal. Its spectrum maps to $U_i{\times }_{U}V$ by the fibre-product description [F3], and the resulting point maps to $v$. Thus the pullback family is jointly surjective.

Let $U_i\to U$ be a declared cover and, for each $i$, let $V_{ij}\to U_i$ be a declared cover. Their composites are étale by [F2]. For any point $u\in U$, joint surjectivity first gives an $i$ and a point $u_i$ over $u$, then the cover of $U_i$ gives a $j$ and a point $v_{ij}$ over $u_i$. The composite image therefore contains every point $u$. The disjoint union of the set-indexed index families is a set, so the composite family is an allowed declared cover. Empty families cause no exception: they cover only the empty object, whose pullbacks along maps into it are empty.

Steps 1.1–1.3 prove the identity, pullback, and composition axioms for a pretopology. Each declared family generates a sieve; the smallest Grothendieck topology containing those sieves and the maximal sieves is obtained by intersecting all Grothendieck topologies containing them. This is the topology specified in [F1], and the pretopology axioms ensure it has the usual covering-family presentation. The proof establishes only the site axioms; it asserts no representability of sheaves or torsors.