The Stacks project

Lemma 59.102.1. With notation as above. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. The rule

\[ (\mathit{Sch}/S)_{ph} \longrightarrow \textit{Sets},\quad (f : X \to S) \longmapsto \Gamma (X, f_{small}^{-1}\mathcal{F}) \]

is a sheaf and a fortiori a sheaf on $(\mathit{Sch}/S)_{\acute{e}tale}$. In fact this sheaf is equal to $\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$ and $\epsilon _ S^{-1}\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_{ph}$.

Proof. The statement about the étale topology is the content of Lemma 59.39.2. To finish the proof it suffices to show that $\pi _ S^{-1}\mathcal{F}$ is a sheaf for the ph topology. By Topologies, Lemma 34.8.15 it suffices to show that given a proper surjective morphism $V \to U$ of schemes over $S$ we have an equalizer diagram

\[ \xymatrix{ (\pi _ S^{-1}\mathcal{F})(U) \ar[r] & (\pi _ S^{-1}\mathcal{F})(V) \ar@<1ex>[r] \ar@<-1ex>[r] & (\pi _ S^{-1}\mathcal{F})(V \times _ U V) } \]

Set $\mathcal{G} = \pi _ S^{-1}\mathcal{F}|_{U_{\acute{e}tale}}$. Consider the commutative diagram

\[ \xymatrix{ V \times _ U V \ar[r] \ar[rd]_ g \ar[d] & V \ar[d]^ f \\ V \ar[r]^ f & U } \]

We have

\[ (\pi _ S^{-1}\mathcal{F})(V) = \Gamma (V, f^{-1}\mathcal{G}) = \Gamma (U, f_*f^{-1}\mathcal{G}) \]

where we use $f_*$ and $f^{-1}$ to denote functorialities between small étale sites. Second, we have

\[ (\pi _ S^{-1}\mathcal{F})(V \times _ U V) = \Gamma (V \times _ U V, g^{-1}\mathcal{G}) = \Gamma (U, g_*g^{-1}\mathcal{G}) \]

The two maps in the equalizer diagram come from the two maps

\[ f_*f^{-1}\mathcal{G} \longrightarrow g_*g^{-1}\mathcal{G} \]

Thus it suffices to prove $\mathcal{G}$ is the equalizer of these two maps of sheaves. Let $\overline{u}$ be a geometric point of $U$. Set $\Omega = \mathcal{G}_{\overline{u}}$. Taking stalks at $\overline{u}$ by Lemma 59.91.4 we obtain the two maps

\[ H^0(V_{\overline{u}}, \underline{\Omega }) \longrightarrow H^0((V \times _ U V)_{\overline{u}}, \underline{\Omega }) = H^0(V_{\overline{u}} \times _{\overline{u}} V_{\overline{u}}, \underline{\Omega }) \]

where $\underline{\Omega }$ indicates the constant sheaf with value $\Omega $. Of course these maps are the pullback by the projection maps. Then it is clear that the sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection, and sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection. The sections in the intersection of the images of these pullback maps are constant on all of $V_{\overline{u}} \times _{\overline{u}} V_{\overline{u}}$, i.e., these come from elements of $\Omega $ as desired. $\square$

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