# restriction and pullback of representable etale sheaf along closed immersion

1. I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my understanding about this.

For simplicity, let's work over a trait $S$, with $s\xrightarrow{i}S\xleftarrow{j}\eta$ the special point and the generic point. Let $G$ be an $S$-scheme, considered as a sheaf on the etale site $(\mathrm{Sch}/S)_{\mathrm{et}}$. Now we have two sheaves associated to $i$.

(1) the restriction of the sheaf $G$ to the site $(\mathrm{Sch}/s)_{\mathrm{et}}$;
(2) the pullback $i^*G$ along the closed immersion $i$

Apparently, the first sheaf is represented by the scheme $G_s=G\times_{S}s$. How about the second one? It seems to me that the second one is represented by $G_s$ if for any scheme $V_s$ over $s$, there exists an 'canonical' ('universal') lifting $V$ over $S$ such that $V\times_{S}s=V_s$ and $Hom_S(V,G)\cong Hom_s(V_s,G_s)$. This rarely happenes.

2. I think a closed subscheme usually doesn't give rise to subsheaf. I used to be very confused with this.

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Are you working with small or big etale sites? Your 1. (1) only makes sense to me in the big etale site, in which case the agreement that you want results from the definition of $i^*G$. – Kestutis Cesnavicius Dec 5 '12 at 11:49
I think one reason for the confusion is the following. The pull-back of a representable étale sheaf generally only coincides with the pull-back (base-change) of the corresponding scheme if the scheme is étale, so in your situation if $G$ is étale over $S$. To see this, remember that the sheaf pull-back is left adjoint to sheaf push-forward; but the push-forward of a representable étale sheaf is not representable in general - unless the representing scheme is étale, in which case the scheme representing the push-forward is a kind of Weil restriction. See the book by Freitag-Kiehl, I,§3,p29. – Damian Rössler Dec 5 '12 at 11:53
(I am working in the small site). – Damian Rössler Dec 5 '12 at 11:54
@ Kestutis Cesnavicius: I think my notation $(Sch/S)_{et}$ is standard for big etale site. – Heer Dec 5 '12 at 14:01
Now I understand why I got confused. For the small etale site, the restriction and pullback don't coincide; but for big etale site, they do. There is a remark in Milne's book 'etale cohomology', chapter II, remark 3.1 (a), which explains the reason in more genearl setting. Thanks @Kestutis Cesnavicius and @Damian Rössler for the comments – Heer Dec 10 '12 at 16:18