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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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