The Stacks project

Lemma 58.96.1. 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 58.39.2. 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 58.87.4 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. $\square$


Comments (0)

There are also:

  • 3 comment(s) on Section 58.96: Comparing ph and étale topologies

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DDW. Beware of the difference between the letter 'O' and the digit '0'.