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It follows from a recent answer 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?

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Would it even be subcanonical? Formally étale does not imply flatness, so at least the usual proof of subcanonicity wouldn't work. –  Daniel Bergh Mar 18 '10 at 8:53
    
I've clarified what I meant. With this, it is automatically subcanonical. The problem that I foresee is that it may not be comparable to the Zariski topology (even in the weakest case I described). Despite this, I wonder if we can still have some notion of algebraic geometry (this should be possible if formally smooth maps satisfy Toen's axioms for a geometric context, but I'm not sure that is the case. –  Harry Gindi Mar 18 '10 at 11:20
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The problem you run into is that you can't do anything interesting. This would be apparent if you focus more effort on understanding the construction of actual non-trivial examples of moduli schemes, algebraic spaces, etc. –  BCnrd Mar 18 '10 at 15:10
    
Do you have any suggestions about where to actually read about these examples (and exercises relating to these constructions)? I don't even know where to start. –  Harry Gindi Mar 18 '10 at 16:24
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Boyarchenko, Dolgachev, Fulton, Lazarsfeld, Mustata, Smith... –  BCnrd Mar 18 '10 at 20:07
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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"

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?
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Since the suggestion to post the 2 ending questions in this answer as its own MO question was wisely deleted, let me revise my comment: please think about those for much longer by yourself. It will be more instructive for you that way. Examples are not hard to find; just think of very simple things. –  BCnrd Mar 18 '10 at 20:31
    
Thank you for your comment. I threw out the questions since it was not a priori clear that the topology in question actually gave something new, and answering these questions would clarify this. I have yet to learn the interrelation between all equivalent definitions of étale maps, so for me it would take much more than 30 seconds to find the answers. No doubt, it will be a good exercise, but for the moment I'd rather spend my effort on understanding non-trivial examples of moduli-schemes and algebraic spaces ;-), which is exactly why I didn't post the questions in a separate thread. –  Daniel Bergh Mar 19 '10 at 10:03
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