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Are they equivalent?

That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If not, what is an example of a sheaf which cannot be extended?

What about for sheaves of abelian groups?

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  • $\begingroup$ the cohomology is the same if it is any help: stacks.math.columbia.edu/tag/03YX $\endgroup$
    – user145520
    Commented Nov 5, 2019 at 1:04
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    $\begingroup$ I'm not sure if extensions always exist, but they're certainly not always unique. For example, there are two natural ways to extend $\Omega_{X/k}$ to the big site: as a quasi-coherent sheaf by $\mathscr F(Y) = (Y\to X)^* \Omega_{X/k}$, or as differentials by $\mathscr G(Y) = \Omega_{Y/k}$. These agree on the small site, but not the big site. This at least shows that the natural morphism $\iota\colon (\mathbf{Sch}/X)_{\operatorname{\acute et}}\to X_{\operatorname{\acute et}}$ does not induce an equivalence on topoi, as $\iota_*\mathscr G\cong\iota_*\mathscr F$ but $\mathscr G \ncong\mathscr F$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 5, 2019 at 4:07
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    $\begingroup$ It does seem to me that $\iota_*\iota^* \cong \operatorname{id}_{X_{\operatorname{\acute et}}}$, so $\iota^* \mathscr F$ is an extension to $(\mathbf{Sch}/X)_{\operatorname{\acute et}}$ of a sheaf $\mathscr F$ on $X_{\operatorname{\acute et}}$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 5, 2019 at 4:10
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    $\begingroup$ Or better, let $F$ be the (big etale sheaf represented by) a point of $X$ i.e. ${\rm Spec}(k)\to X$, then the restriction of $F$ to the small etale site is usually the empty sheaf. $\endgroup$
    – Piotr Achinger
    Commented Nov 5, 2019 at 15:06

1 Answer 1

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There is a site morphism $$i: X_{\rm \acute{e}t}\to {\rm \acute{E}t}(X),$$ giving an adjunction (indeed, a geometric morphism of topoi—abstract nonsense) $${\rm EXT}=i^*: \mathsf{Shv}(X_{\rm \acute{e}t})\rightleftarrows\mathsf{Shv}({\rm \acute{E}t}(X)): i_*={\rm Res},$$ where ${\rm EXT}=i^*$ is given by some Kan extension, $i_*={\rm Res}$ is really the restriction.

Standard text books on étale cohomology say that the unit $F\to{\rm Res}({\rm EXT}(F))$ of the adjunction $({\rm EXT}, {\rm Res})$ is an isomorphism.

So every sheaf on the small étale site is always extendable to the big étale site and restricts back to the original sheaf. So your question is not a correct one.

On the other hand, the counit ${\rm EXT}({\rm Res}(G))\to G$ is usually not an isomorphism. As in the above comments, different big étale sheaves can restrict to isomorphic small étale sheaves. The correct question is to ask for such examples, already given by the above comments.

What is important in étale cohomology theory is, for any $A\in\mathsf{ShvAb}(X_{\rm \acute{e}t}), B\in\mathsf{ShvAb}({\rm \acute{E}t}(X)), U\in X_{\rm \acute{e}t}$, there are canonical isomorphisms $${\rm H}^n_{\rm \acute{e}t}(X; A)\cong{\rm H}^n_{{\rm \acute{E}t}(X)}(X; {\rm EXT}(A)), {\rm H}^n_{\rm \acute{e}t}(X; {\rm Res}(B))\cong{\rm H}_{{\rm \acute{E}t}(X)}^n(X; B); $$ $${\rm H}^n_{\rm \acute{e}t}(U; A)\cong{\rm H}^n_{{\rm \acute{E}t}(X)}(U; {\rm EXT}(A)).$$

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