Let i : Y —> X be a closed immersion of schemes. This gives rise to a morphism of topoi i_{big} : (Sch/Y)_{fppf} —> (Sch/X)_{fppf}. Question: Is the direct image functor i_{big, *} is exact on the category of abelian sheaves?

My guess is no. To find an example we can look for an Artinian local ring A with an ideal I and a finite flat local ring map A/I —> C such that there does not exist any finite flat ring map A —> B with the property that A/I —> B/IB factors through C. Namely, in this case the map of abelian sheaves

(Z/2Z)_{Spec(C)} —> Z/2Z

on Y = Spec(A/I) is fppf surjective because {Spec(C) —> Spec(A/I)} is an fppf covering. Here the first sheaf is the free Z/2Z-module on the fppf sheaf represented by Spec(C) over Y. But

i_*((Z/2Z)_{Spec(C)}) —-> i_*(Z/2Z)

is not surjective since the section 1 does not lift fppf locally on X = Spec(A) by our assumption on A/I —> C. To make an explicit example you probably can do something similar to Exercise Tag 02CV but I haven’t quite been able to make it work yet. Leave a comment if you have an example, or a reference, or if you think the answer to the question is yes.