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?<\/p>\n
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<\/p>\n
(Z\/2Z)_{Spec(C)} —> Z\/2Z<\/p>\n
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<\/p>\n
i_*((Z\/2Z)_{Spec(C)}) —-> i_*(Z\/2Z)<\/p>\n
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<\/a> 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.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading