The Stacks project

Remark 59.48.7. In Lemma 59.48.6 the case $\tau = fppf$ is missing. The reason is that given a ring $A$, an ideal $I$ and a faithfully flat, finitely presented ring map $A/I \to \overline{B}$, there is no reason to think that one can find any flat finitely presented ring map $A \to B$ with $B/IB \not= 0$ such that $A/I \to B/IB$ factors through $\overline{B}$. Hence the proof of Lemma 59.48.5 does not work for the fppf topology. In fact it is likely false that $f_{big, *} : \textit{Ab}((\mathit{Sch}/X)_{fppf}) \to \textit{Ab}((\mathit{Sch}/Y)_{fppf})$ is exact when $f$ is a closed immersion. If you know an example, please email stacks.project@gmail.com.