Hi!


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?



Essential geometric morphisms on the étale site. 
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.


