I noticed that in a lecture at the IHES, Peter Scholze seemed to refer to a "mixed-characteristic field". Now, I know what a mixed characteristic ring is – it's a ring $A$ such that the localizations at primes $\kappa(\mathfrak{p})$ vary from being characteristic $0$ to being characteristic $p>0$. However, with this definition, it makes no sense to talk about a "mixed-characteristic field".

It seems like he is considering something of the sort $\mathbb{Q}_p(x)$ to be a mixed characteristic field, but I really can't see any intrinsic property of a field that fits the idea of "mixed-characteristic".

  • $\begingroup$ It's basically a field which maintains itself characteristic 0, while its quotient ring has a positive characteristic $\endgroup$ – enedil Apr 24 '17 at 23:18
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    $\begingroup$ You might want to read Scholze's answer detailing some of fis research: mathoverflow.net/a/66563/62132 $\endgroup$ – enedil Apr 24 '17 at 23:24

In the context where Scolze is using this term, such a fields comes with a valuation. The corresponding valuation ring has a residue field (this is the quotient by its unique maximal ideal). If the characteristic of this residue field is different from that one of the initial field, then one says that the initial field is of mixed characteristic. Note that this can only happen if the initial field has characteristic 0.

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