2
$\begingroup$

Consider a scheme $X$. Grothendieck, SGA 4, Exposé VIII, Theorem 3.5, says that the family of its geometric points is conservative, i.e. isomorphisms of sheaves on $X$ can be "checked" on the pullback via the geometric points.

To be precise, he doesn't speak of pullbacks, but of fiber functors, that are just the composition of the pullback with the global sections on the geometric point. But just before he had defined such fiber functor (only) for sheaves on the étale site; while here $F$ and $G$ are sheaves on $X$. This "confusion" would seem to suggest that, for him, sheaves on $X$ and sheaves on the étale topos are related.

Actually, an object in the étale topos yields an open set of $X$, by taking its image. (This corresponds to seeing the étale site $X_{ét}$ immersed as a site into $\mathcal O(X)$, the site of open sets of $X$.) So if I have a sheaf on $X$ I can recover a sheaf on the étale site.

But conversely, if I have a sheaf $F$ on the étale site, is there a way to see it as a sheaf $F'$ on $X$ in order to apply the theorem? Perhaps $F'(U), U\subset X$, could just be defined by glueing data on an open cover of $U$ made by étale schemes?

Or am I just missing something obvious?

$\endgroup$
  • 1
    $\begingroup$ A sheaf on the etale site induces one on the Zariski site. Let $\mathcal{F}$ be a sheaf on the etale site of $X$. Every open inclusion $U \hookrightarrow X$ it etale, and allows us to define $\mathcal{F}_{\text{Zar}}(U) = \mathcal{F}(U \hookrightarrow X)$. $\endgroup$ – RghtHndSd Jul 1 '18 at 22:30
2
$\begingroup$

"Sheaf on $X$" here means a sheaf on the étale site of $X$, not a sheaf on the open sets of $X$. This convention is stated in section 1.3 of Exposé VII.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.