Example 59.15.7. Consider the presheaf
\[ \begin{matrix} \mathcal{F} :
& (\mathit{Sch}/S)^{opp}
& \longrightarrow
& \textit{Ab}
\\ & T/S
& \longmapsto
& \Gamma (T, \Omega _{T/S}).
\end{matrix} \]
The compatibility of differentials with localization implies that $\mathcal{F}$ is a sheaf on the Zariski site. However, it does not satisfy the sheaf condition for the fpqc topology. Namely, consider the case $S = \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$ and the morphism
\[ \varphi : V = \mathop{\mathrm{Spec}}(\mathbf{F}_ p[v]) \to U = \mathop{\mathrm{Spec}}(\mathbf{F}_ p[u]) \]
given by mapping $u$ to $v^ p$. The family $\{ \varphi \} $ is an fpqc covering, yet the restriction mapping $\mathcal{F}(U) \to \mathcal{F}(V)$ sends the generator $\text{d}u$ to $\text{d}(v^ p) = 0$, so it is the zero map, and the diagram
\[ \xymatrix{ \mathcal{F}(U) \ar[r]^{0} & \mathcal{F}(V) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(V \times _ U V) } \]
is not an equalizer. We will see later that $\mathcal{F}$ does in fact give rise to a sheaf on the étale and smooth sites.
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