# Rigid analytic geometry in characterstic 0 vs positive characteristic

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?

• a) I thought, resolution of singularities was not known to be true only in char 0. b) What do you mean by "vice-versa"? – Helene Sigloch Apr 7 '16 at 10:02
• @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. – Andrei Halanay Apr 7 '16 at 14:35
• 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. – Helene Sigloch Apr 8 '16 at 7:56

## 1 Answer

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.