The Stacks project

Example 59.15.7. Consider the presheaf

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

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

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.