Let $X$ be a noetherian scheme separated and of finite type over a noetherian base $S$ and $\epsilon$ a section of its structure morphism. I want to consider $S$ via this section as a closed subscheme of $X$.
Given a coherent sheaf $F$ on $X$ with the property $supp(F)=S$ (here consider $S$ embedded in $X$), where $supp$ denotes the support of a sheaf, then my question is:
Does one find a surjective module homomorphism
$F \rightarrow \epsilon_*\mathcal O_S$?
It does not need to be canonical; e.g. if $S$ is the spectrum of a field $k$ one certainly does, by Adjunction and simply chosing a projection of the $k-$vector space $F/\mathcal m$ to $k$, where $\mathcal m$ is the maximal ideal of the local ring of $\epsilon(pt)$, with $pt$ the point of $Spec(k)$.
I thought a pretty while about it, but don't see how one would get the epimorphism in the general case in question, so my guess would be no, in general. But maybe some additional properties I don't see at the moment would improve the situation.