# Fiberwise acyclic, locally acyclic morphisms

The quick definition of a map $f \colon X \to B$ of schemes being acyclic is that the natural unit of adjunction $\def\id{\operatorname{id}}\id \to f_* f^*$ is an isomorphism, where we take $f_*$ to mean the derived pushforward, and I'm using etale sheaves. The one class of morphisms I'm aware of being acyclic are relative affine spaces, my source for all this being SGA4 xv and SGA 4 1/2. In this instance these morphisms are actually universally acyclic (i.e. acyclic after arbitrary base change) and although Artin makes a comment in SGA to the effect that he doesn't know whether this is necessarily always true, I rather doubt it (if there's anything to be said there, though, please say it).

Suppose I happen to have another acyclic morphism, though, and suppose I wish to show that it is universally so. For proper morphisms this is easy: by proper base change and checking an isomorphism on the stalks, acyclicity is equivalent to fiberwise acyclicity (that is, acyclicity after base change to any geometric point), and this is definitely a universal property.

What about for smooth morphisms? Every smooth morphism is universally locally acyclic (for coefficients prime to the characteristic), and using smooth base change, acyclicity implies fiberwise acyclicity. These are both universal properties, but do they together imply global acyclicity?

Ultimately, of course, I would like to know any criterion that can be piled on top of "smooth acyclic" to imply "universally acyclic".

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