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There is a risk that this question might be utterly trivial, and if it is, my sincerest apologies, but I haven't been able to find anything in the literature.

Let $f:X \rightarrow Y$ be an étale morphism of schemes. This gives rise to a geometric morphism of topoi $f^\ast:Sh(Y) \rightleftarrows Sh(X) : f_\ast$, and my question is now: In what cases do $f^\ast$ have a left adjoint? Are there any explicit descriptions for this for say affine schemes?

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I would mostly be interested in the case for affine schemes. Spec Z would be of interest, and some nice restrictions on X is OK. More generally, I am curious what I should look for to think if it is plausible that an adjoint exists. – Dedalus Feb 28 '13 at 18:20
So $Sh(X)$ means sheaves on the small étale site? I only ask because it is stated that $f$ is étale in the question, but the site is only mentioned in the subject line. – Matt Feb 28 '13 at 18:55
Thanks, yes , I mean the small étale site! – Dedalus Feb 28 '13 at 19:28
up vote 9 down vote accepted

The answer is always. Let $\mathrm{Et}(X)$ and $\mathrm{Et}(Y)$ denote the étale sites. There is a functor $f_! : \mathrm{Et}(X) \rightarrow \mathrm{Et}(Y)$ sending an étale $X$-scheme $p : U \rightarrow X$ to $f \circ p : U \rightarrow Y$. This functor is cocontinuous (SGA4.III.2.1) and continuous (SGA4.III.1.1). By SGA4.III.2.6, any functor that is both continuous and cocontinuous gives rise to a morphism of topoi $(f^\ast, f_\ast)$ where $f^\ast$ has a left adjoint.

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Here's another way of seeing this which some people might like. Let $F$ be a sheaf on $X$. Then $F$ is represented by an algebraic space $Z$ that is etale over $X$. (For this to be true in general, we must allow $Z$ to be non-separated and even non-quasi-separated.) Then $Z$ can be viewed as an etale algebraic space over $Y$, via the map $f$. The sheaf it represents is $f_!(F)$. – JBorger Mar 5 '13 at 12:34

As Jonathan said, the answer is always, and the left adjoint $f_!$ has a simple description as the "extension by $\emptyset$" functor. Think of an open immersion $j:U\rightarrow X$ of topological spaces, then we have the usual extension by $\emptyset$ functor by sheafififying the presheaf

$j_!\mathcal{F}(W)=\mathcal{F}(W)$ if $W\subset U$

$j_!\mathcal{F}(W)=\emptyset$ otherwise.

It's an easy calculation to show that this is left adjoint to $j^*$.

Translating this to the étale topology, if $f:X\rightarrow Y$ is an étale morphism of schemes, we have the extension by $\emptyset$ functor by sheafififying the presheaf

$f_!\mathcal{F}(W\overset{f}{\rightarrow} Y)=\coprod_{g\in\mathrm{Hom}_Y(W,X)}\mathcal{F}(W\overset{g}{\rightarrow} X)$

and again it's fairly easy to verify that $f_!$ does what we want it to.

It's worth noting that to get an adjoint for abelian sheaves, we need to replace this coproduct by the coproduct in the category of abelian sheaves, i.e. direct sum. Thus $f_!\mathcal{F}$ depends on whether we are considering $\mathcal{F}$ as a sheaf of sets or of abelian groups.

It's possibly also worth noting that we don't get a geometric morphism $f^*:\mathrm{Sh}(Y)\leftrightarrows \mathrm{Sh}(X):f_!$ because $f_!$ doesn't preserve limits. Despite this, the version of $f_!$ for abelian sheaves is actually exact!

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