Let $f: X\rightarrow S$ be a quasi-compact, quasi-separated, flat morphism of schemes, with $S$ locally Noetherian. Also, fix an integer $i\geq 1$.

**Question 1:** Does $H^i(X_s,\mathcal{O}_{X_s})=0$, for all $s\in S$, imply that $R^if_\ast\mathcal{O}_X=0$?

The answer is *yes*, if $f$ is also assumed to be *proper* (flat base change+theorem of formal functions+easy direct proof for $S$ Artinian local). In general, I would expect that there are some simple counterexamples but, to my surprise, I didn't find anything in the literature - probably I simply looked in the wrong places! I should also add that I would be particularly interested in counterexamples in characteristic zero, i.e. where $f$ is a morphism of schemes over a field of characteristic zero.

Here is a variant of question 1:

**Question 2:** Assume in addition that $f$ has a section $e: S\rightarrow X$. What is the answer to question 1 then?

The "raison d'être" for asking question 2 in addition is that a colleague explained to me how, granting existence of a section, the assumptions above ensure that $H^0(X_s,\mathcal{O}_{X_s})=\kappa(s)$, for all $s\in S$, implies that $f_\ast\mathcal{O}_X=\mathcal{O}_S$. So, at least in this very special case, the fibres of $f$ "remember" what the global sections of $f$ are.