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    $\begingroup$ This is a good idea, but I'm not convinced that $X^s$ is a scheme. Infinite products and inverse limits of non-affine schemes can be rather thorny. $\endgroup$ Commented Jul 29, 2013 at 4:10
  • $\begingroup$ Thorny or nonexistent? $\endgroup$ Commented Jul 29, 2013 at 4:18
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    $\begingroup$ Inverse limits of arbitrary diagrams don't exist in the category of schemes. How does the argument deal with that? At least it seems to show that $X^s$ exists as a pro-scheme... $\endgroup$ Commented Jul 29, 2013 at 4:19
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    $\begingroup$ mathoverflow.net/questions/65506/… $\endgroup$ Commented Jul 29, 2013 at 4:22
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    $\begingroup$ Yes, this is of course the obvious flaw. But at least we can see that any counterexample must be pathological in some exciting new way, namely that it has infinitely many incompatible maps to separated schemes. $\endgroup$ Commented Jul 29, 2013 at 4:54