@zzy Obviously, étale cohomology over non-separably closed fields is not a Weil cohomology. Over arbitrary fields of characteristic zero, algebraic de Rham cohomology is. Over rings of integers, integral algebraic de Rham cohomology even puts an integral structure on it, that specializes to crystalline cohomology of every closed fiber. Algebraic de Rham cohomology and crystalline cohomology are non-geometric Weil cohomologies, in the sense that they are Weil cohomologies for (smooth proper) varieties defined over possibly non-separably closed fields.

@zzy I’m not sure why one would be interested in Weil cohomologies abstractly, unless one has a model to run calculations and arguments of geometric flavor (such as interaction with specialization maps). Also, you talk about the proper base change theorem by stressing the words “base change” as if I was talking about cohomology over non-separably closed (or non field) bases, but the proper (and smooth) base change theorem in $\ell$-adic cohomology is about geometric cohomology (i.e. over separably closed fields).

@zzy I mentioned most of the currently known Weil cohomologies, and argued that for smooth projective varieties over separably closed fields your question has a positive answer for all of them. If you ask about an arbitrary Weil cohomology, again for smooth projective varieties, then the answer is known (and positive) only over finite fields by Katz-Messing. If you’re asking if an analog of Katz-Messing is known over arbitrary separably closed fields (again, for smooth projective varieties), the answer is no: it should follow from a combination of the standard conjectures

In the special case when $\text{Spec}(A)$ is an affine space over $R/I$, or a distinguished affine open of a projective space over $R/I$, can one lift the relations directly, to get a surjective lift $A\to B$ on the nose without replacing $\text{Spec}(A)$ with an open neighborhood of $\text{Spec}(A_0)$? I’d just like to be sure

@DavidRoberts Let’s say $M$ is connected. But yes I would content myself of a cover made my opens that are finite disjoint unions of contractibles, all whose multiple intersections are also finite disjoint unions of contractibles.