Here are some questions of the form: I don’t know this, do you? These questions showed up in the stacks project, but we did not take a lot of time to find them in the literature or try to solve them. Maybe you know how to do some of these or a reference? Here they are:

  1. Does there exist a scheme which is connected, all of whose local rings are domains, but which is not irreducible?
  2. Let X be a scheme over a field k. Let k ⊂ K be an extension of fields. Let T be a connected component of X_K. Is the image of T in X_k a connected component of X?
  3. Is it true that a Noetherian ring all of whose local rings are Japanese is Japanese?
  4. If f : Z —> X is a closed immersion of schemes, then is f_* exact on the category of abelian sheaves on (Sch/Z)_{fppf}? I think probably not in general, but I don’t have an example. Do you?
  5. If X is an algebraic space (as defined in the stacks project, so not necessarily quasi-separated) does X satisfy the sheaf condition for fpqc coverings?

PS: Question 5 has a positive answer for quasi-separated spaces, see Lemma Tag 03WB. David Rydh has some more results on this. Maybe the real question here is if one can make some horrible counter example, or whether there is some straightforward argument covering all algebraic spaces that everybody has missed so far.

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