It follows from a [recent answer](https://mathoverflow.net/questions/16132/formally-etale-at-all-primes-does-not-imply-formally-etale/17775#17775) that even when a ring is formally étale rather than étale, we can check this condition on localizations, and hence stalks. It's not hard to show that we can define a "formally étale" topology on $Aff$. Presumably there's a good reason for requiring finite presentation), but I can't think of a reason why off of the top of my head. 


Specifically, we let the covering families be jointly faithfully flat with each morphism formally étale. This is by construction subcanonical. 


Questions: Why are finiteness conditions necessary for schemes in general and étale maps in particular. 


What kinds of problems will we run into if we do not put finitness conditions on the "formally étale" topology?


If this topology fails in some way, can requiring that covers are finite families of morphisms (quasicompact), or that the morphisms in the cover are themselves flat, or even both? This would give us a topology similar to the fpqc topology, except in that all covering families would be made up of formally étale morphisms. The only difference between this topology and the étale topology is the finite presentation of the morphisms in the covering families. Does this still not work?



Why do we need finiteness conditions for formally étale morphisms?
I guess, to get a good theory, you would want to be able to have effective descent for certain classes maps. In Brian Conrads answer to this question:


"[Quasi-separatedness for Algebraic Spaces](https://mathoverflow.net/questions/16381/quasi-separatedness-for-algebraic-spaces)"


he gives references to how you can work with algebraic spaces without separated assumptions. The cruical step in proving that étale equivalence relations give schematic quotient maps, involves descending étale separated maps. This is possible (in the affine case) for faithfully flat, universally open maps (i.e étale or fppf). Even if you set up your topology such that it lies between the fpqc and Zariski topologies, it is not immediate that this would work.


By the way, the only counterexamples I have seen for formaly étale maps not being étale, involves non flat maps. This leads to two questions:


* Are there flat, formally étale maps (between affines) which aren't fppf?
* Are there flat, formally étale, universally open maps (between affines) which aren't fppf?


