Let $f: X \to Y$ be a morphism between two irreducible schemes and $\mathcal{F}$ sheaf on the small étale site $Y_{ét}$. My question is more or less "dual" to this one:
Question: Under which "reasonable" (=not too "exotic") assumptions do we know something about surjectivity or injectivity of the unit map under $(f^*,f_*)$ adjunction
$$ \mathcal{F} \to f_*f^*\mathcal{F}?\tag{$*$}\label{461816star}$$
If the question is too broad / unconcrete, let's pick the following "usual" conditions I'm principally interested in: It seems to be reasonable to ask for conditions posed on the level of the morphism $f$ (e.g., if $f$ is (open, closed) immersion, affine,.… etc + maybe something on the schemes $X$, $Y$ themselves) (I)
and on the level of the sheaf $\mathcal{F}$ (e.g., assuming it to be (étale) locally constant, locally free, etc.) (II).
Clearly the question can be discussed on the level of stalks, so let $\overline{x}$ be a geometric point and $\mathcal{F}_{\overline{x}}= \varinjlim_{(U,u) \text{ étale nbhds of } x} \mathcal{F}(U)$ and
the question becomes if the stalk at $f_*f^*\mathcal{F} $ is going to be exhausted by this inductive system.
On assumptions of type (I). I think that one can literally copy the proof for Zariski site to see that if $f$ is a closed immersion the unit map \eqref{461816star} is an isomorphism, for open immersion it is an isomorphism in points lying in the image, but not necessarily for points outside the image of $f$. Do we know more thinking in this direction?
What about the situation $X= \eta_Y$ the generic point of $Y$ and $f$ the canonical inclusion? Can we for example expect that \eqref{461816star} is an isomorphism if we require additionally something on $\mathcal{F}$, e.g., étale locally constant?
What about affineness of $f$ + again additional assumptions on $\mathcal{F}$, e.g., again étale locally constant?
It seems that the argument here by Piotr Achinger for counit cannot be imitated because even though the composition $ f^*\mathcal{F} \to f^* f_* f^* \mathcal{F} \to f^* \mathcal{F} $ is the identity, we cannot "draw back" information about \eqref{461816star}, or am I missing something?
Best Answer
Two observations:
(1) This unit is injective if $f$ is set-theoretically surjective: For the image of a section nonvanishing at a point $x$ to vanish, the associated section of $f^* \mathcal F$ on the inverse image open set would have to vanish, so it would vanish at every point, including the inverse image of that point, which is impossible.
(2) This unit is surjective if $f$ is proper with geometrically connected fibers: Proper base change lets us reduce to the case where the base is a geometric point, in which case $\mathcal F$ must be a constant sheaf, and then this is the computation of the global sections of the constant sheaf on connected spaces. This generalizes the closed immersion case you mention.