5
$\begingroup$

This question is motivated purely by curiosity. In algebraic geometry there is a major distinction between the world of characteristic $0$ and that of characteristic $p > 0$ with different methods, different results available etc.

From reading a number of books and papers I got the idea that in the case of rigid analytic geometry the distinction between the two worlds is not that important. So my question is:

Are there results in rigid geometry that are only known to be true in characteristic $0$ (like Hironaka desingularisation in algebraic geometry), or vice-versa, results that are only known to be true in positive characteristic?

$\endgroup$
3
  • $\begingroup$ a) I thought, resolution of singularities was not known to be true only in char 0. b) What do you mean by "vice-versa"? $\endgroup$ Apr 7, 2016 at 10:02
  • $\begingroup$ @HeleneSigloch a)As far as I am aware in there is no known resolution of singularities in positive characteristic and dimension greater then 4. b) I was asking for results true (or proven) only in positive characteristic. $\endgroup$ Apr 7, 2016 at 14:35
  • $\begingroup$ b) Ah, ok. I don't know any. a) Yes, that's what I thought. I hope it is ok if I edit your question. Right now it can be read as if resolution of singularities was known to be false in positive characteristic. $\endgroup$ Apr 8, 2016 at 7:56

1 Answer 1

5
$\begingroup$

Resolution of singularities for rigid analytic varieties of equal characteristic zero follows from resolution of singularities for schemes of characteristic zero (Nicaise, A trace formula for rigid analytic varieties etc., 2009, Proposition 2.43).

There are more examples where the characteristic plays a role, e.g. in Van der Put, Cohomology on affinoid spaces, 1982. Here the reason is the radius of convergence of the logarithm.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.