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I am wondering whether the analogue of III.1.27 in Milne's "Etale cohomology" holds true if one works with fppf cohomology with supports instead of etale cohomology with supports. More precisely, let $f\colon X^{\prime} \rightarrow X$ be fppf, let $Z \hookrightarrow X$ be a closed subscheme, and let $Z^{\prime} \hookrightarrow X^{\prime}$ be its preimage, i.e., $Z^{\prime} = Z \times_X X^{\prime}$. Suppose that $f|_{Z^{\prime}}\colon Z^{\prime} \rightarrow Z$ is an isomorphism and also that $f(X^{\prime} - Z^{\prime}) \subset X - Z$. Let $F$ be a sheaf on $X_{fppf}$. Is it true that the natural morphism of fppf cohomology groups with supports $H^i_{Z}(X, F) \rightarrow H^i_{Z^{\prime}}(X^{\prime}, f^*F)$ is an isomorphism?

As in loc. cit., one is reduced to considering $i = 0$, in which case the proof of injectivity continues to work. Imitating the proof of surjectivity in loc. cit., one would take $\gamma^{\prime} \in \Gamma_{Z^{\prime}}(X^{\prime}, F) \subset \Gamma(X^{\prime}, F)$ and would attempt to glue it with the zero section on $X - Z$. To check the glueing condition, one would need to check in particular that the two pull-backs of $\gamma^{\prime}$ to $X^{\prime} \times_X X^{\prime}$ agree. This step is not explained explicitly in loc. cit., presumably because it is clear in the view of II.3.11. The latter says that the push-forward establishes an equivalence of the category of etale sheaves on $Z$ onto the full subcategory of etale sheaves on $X$ consisting of those sheaves that are supported on $Z$. That brings me to my second question: is there an analogue of this result for the fppf topology?

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Assume X is noetherian. The hypothesis says that $\Omega^1_f$, whose formation commutes with base change, has vanishing fibres along all points in $Z'$. By Nakayama, $\Omega^1_f$ is supported on a closed subset that misses $Z'$ entirely. The \'etale locus $U \subset X$ of $f$ then contains $Z$. Since the cohomology supported along $Z$ is insensitive to passing to opens containing $Z$, we may replace $X$ with $U$ (and $X'$ by the preimage) to assume $f$ is \'etale. Now I didn't check this, but I believe the argument in Milne works for fppf cohomology. – anon Feb 21 '13 at 17:14
Thanks for your comment! As you explain, in the Noetherian case one can replace $X^{\prime}$ by an open $V \subset X^{\prime}$ containing $Z^{\prime}$ and replace $X$ by its image $U = f(V)$ to reduce to the case where $f$ is etale. In this case, for the purpose of showing that the two pull-backs of $\gamma^{\prime}$ to $X^{\prime} \times_X X^{\prime}$ agree, one can restrict the attention to the small etale site, which is the case treated in Milne. – Kestutis Cesnavicius Feb 21 '13 at 19:56
I don't think the second question has a positive answer. Take $X$ to be the spectrum of a dvr with uniformizer $t$, and set $Z \subset X$ to be the closed point defined by $t = 0$. Then the "kernel of multiplication of $t$" defines a sheaf F on the big fppf site of X-schemes. This sheaf also vanishes when restricted to the compliment of Z (I take this to mean "supported on Z"). However, I don't think $F$ is the pushforward of $F|_Z$ (by computing global sections). – anon Feb 22 '13 at 2:43

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