Let \(S\) be a scheme and \(\Sch_{/S}\) the category of \(S\)-schemes.
(Étale and fppf topologies)
A family of \(S\)-morphisms \(\{U_i \to U\}_i\) in \(\Sch_{/S}\) is an
étale covering if each \(U_i \to U\) is étale and \(\bigsqcup_i U_i \to U\) is surjective;
fppf covering if each \(U_i \to U\) is flat and locally of finite presentation and \(\bigsqcup_i U_i \to U\) is surjective.
With these coverings \(\Sch_{/S}\) is the big étale site \((\Sch_{/S})_{\mathrm{et}}\), respectively the big fppf site \((\Sch_{/S})_{\mathrm{fppf}}\).
(Small étale site)
The small étale site \(S_{\mathrm{et}}\) of \(S\) is the full subcategory of \(\Sch_{/S}\) whose objects are the étale morphisms \(U \to S\), with the étale coverings as coverings.
Every Zariski open covering is an étale covering, and every étale covering is an fppf covering, because étale morphisms are flat and locally of finite presentation. So every fppf sheaf is an étale sheaf, and every étale sheaf restricts to a Zariski sheaf on each \(S\)-scheme.
(Representable functors are sheaves)
For every \(S\)-scheme \(X\), the functor \(h_X = \Hom_S(-, X)\) is a sheaf on \((\Sch_{/S})_{\mathrm{fppf}}\), hence on \((\Sch_{/S})_{\mathrm{et}}\). For every quasicoherent \(\OO_S\)-module \(\mathcal{M}\), the functor \((f \colon U \to S) \mapsto \Gamma(U, f^* \mathcal{M})\) is an fppf sheaf.
Let \(n \geq 1\) be invertible on \(S\). On \(S_{\mathrm{et}}\), the multiplicative group \(\GG_m \colon U \mapsto \Gamma(U, \OO_U)^\times\) and its subgroup \(\mu_n\) of \(n\)-th roots of unity fit into the Kummer sequence \[1 \to \mu_n \to \GG_m \xrightarrow{\, u \mapsto u^n \,} \GG_m \to 1,\] which is exact on \(S_{\mathrm{et}}\): every unit \(u\) acquires an \(n\)-th root \(z\) on the étale covering \(\Spec_U \OO_U[z]/(z^n - u) \to U\). The sequence is not exact for the Zariski topology. For \(S = U = \Spec \CC[t, t^{-1}]\) and \(n = 2\), the unit \(t\) is not a square on any nonempty Zariski open subset of \(U\).
Dependencies
D-MORFLATFlat morphismsusesD-GROTHTOPSites, sheaves on a site, and topoiusesD-MORETALEÉtale morphismsuses
Backlinks
T-FFDESCENTFaithfully flat descent and the Amitsur complexrelated-toD-ALGSPACEÉtale equivalence relations and algebraic spacesuses