I'm working on problem 5.2.16 in Liu's Algebraic Geometry and Arithmetic Curves.
Let $f : X \to Y$ be an affine morphism and $\mathcal{F}$ a quasi-coherent sheaf on $X$. We would like to show the canonical homomorphism $$f^*f_*\mathcal{F} \to \mathcal{F}$$ is surjective.
We want to show the stalk map at $x$ is surjective for all $x \in X$: $$(f^*f_*\mathcal{F})_* \cong f_*\mathcal{F}_{f(x)} \otimes_{\mathcal{O}_{Y,f(x)}} \mathcal{O}_{X,x} \to \mathcal{F}_x.$$ I know in general $f_*\mathcal{F}_{f(x)}$ is not isomorphic to $\mathcal{F}_x$, but I'm not sure how they are related. Could someone give me a hint on how to approach this? Thanks!
We may assume that $Y=\operatorname {Spec}A$ is affine.
Then $X$ will be affine too, $X=\operatorname {Spec}B$, and $f$ will correspond to a ring morphism $\phi:A\to B$.
The quasi-coherent sheaf $\mathcal{F}=\tilde N$ will be associated to some $B$-module $N$ and the canonical map $f^*f_*\mathcal{F} \to \mathcal{F}$ corresponds to the canonical $B$-module morphism $$s:{_AN}\otimes_AB\to N:n\otimes b\mapsto bn$$ where $_AN$ is the $A$-module obtained from $N$ by the restriction of scalars $\phi:A\to B$.
Since $s(n\otimes 1_B)=n$, $s$ is surjective and thus the required canonical map $$\tilde s:f^*f_*\mathcal{F} \to \mathcal{F}$$ is surjective too by exactness of the functor quasicoherentification™