@EricPeterson What do you mean by "alternating powers don't exist" exactly? I mean what kind of universal property? Somehow I had the impression that once you have the box tensor product you can define alternating powers by existence of colimits, maybe that doesn't have a good universal property? Regarding the previous comment, I didn't quite get what's the problem you have in mind.

I could imagine there being a problem with the fact that flatness would imply dimension is constant which would then imply that some etale components should pop up at lower heights (because dependence between height and dimension for p-divisible groups). Similarly to how mu_p^infty becomes etale rationally. Maybe that gives a contradiction somehow? Or were you trying to say something else?