Let $k$ be a closed field of characteristic $p \geqq 0$ and $\Lambda = \mathbf{Z}/\ell$, $\ell \neq p$. Recall that a morphism $f \colon X \to S$ of $k$-varieties is said to be locally acyclic if for all points $x \in X$ and geoemetric specializations $t \to \operatorname{Spec} \mathcal{O}_{S,s}^\texttt{sh}$ the "variety" of vanishing cycles $$X^x_t = \operatorname{Spec} \mathcal{O}_{X,x}^\texttt{sh} \times_{\operatorname{Spec} \mathcal{O}_{S,s}^\texttt{sh}} t$$ is acyclic (in the sense that $\Lambda \xrightarrow{\sim} R\Gamma(X^x_t, \Lambda)$.) Examples of locally acyclic morphisms abound. Here are the main sources of these:
- Smooth morphisms are locally acyclic
- By Gabber's theorem, if $X \to S$ is locally acyclic then so is $X \times T \to T$ for any $T \to S$.
- Any morphism $X \to \operatorname{Spec} k$ is locally acyclic
- Anything locally on source and target of the form above
Locally acyclic morphisms play a role in the étale topology of schemes akin to flat morphisms for the zariski topology (compare: locally acyclic base change and flat base change). Note also that every morphism in the list above is also flat! This begs the question:
Is there a locally acyclic morphisms of $k$-varieties which is not flat?
EDIT: As Jason Starr pointed out in the comments, it is easy to find counterexamples unless one also asks for reducedness. One natural condition one could impose is normality, or even smoothness, but I will leave it open ended for now as I am still not sure what is the most natural setting for this question.
Here is some positive results I came up upon further thinking: if $f \colon Y \to X$ is locally acyclic and proper, $X$ is regular and $Y$ is CM then $f$ is flat by proper-locally acyclic base change and miracle flatness. This gives me hope that we can maybe prove some result for smooth $k$-varieties (a related question here, perhaps being which LA morphisms can be "compactified" to an LA proper morphism.)
