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the entry valuation would deserve more clarification on that issue alluded to under “Sometimes one also…” and where the min-style definition appears the max-style definition should also appear.
The entry should say that at least with some qualification added, then a valued field is a normed field with multiplicative norm. – Or should it be semi-normed?
I could fiddle with it, but I feel I don’t quite get why the terminology here is so non-uniform that I am afraid I am missing something and maybe a more expert person should help.
In Scholze 11, remark 2.3 is a useful comment:
The term valuation is somewhat unfortunate: If $\Gamma = \mathbb{R}_{\geq 0}$, then this would usually be called a seminorm, and the term valuation would be used for (a constant multiple of) the map $x \mapsto - log {\vert x \vert}$. On the other hand, the term higher-rank norm is much less commonly used than the term higher-rank valuation.
Ah, now I remember that the entry absolute value was supposed to take care of this.
We need to think about what to do here. The entry absolute value declares that it discusses what is commonly called just “valuation”, but that convention is hard to maintain throughout the nLab.
I think we should merge the entries “valuation” and “absolute value” to a single entry that starts out clearly saying that there is a more restrictive and a more general sense of the term “valuation”, then first discusses the more restrictive sense and then the more general sense.
Or something like this. Does anyone feel like doing the required clean-up here?
I would keep two entries
there is also more than one notion of absolute value, and not all such notions are commonly called valuations…
it is not only that valuation can be more restrictive and less restrictive but that two different notions are called valuation just because roughly exponentiation can make one into another (but only in a subset of cases).
the absolute value entry is in quite good condition and I would not like to mess it with valuation entry
You call wikipedia succinct as it chose only one of the two kinds.Neglecting complexity makes things simpler isn’t it ? I prefer that the explicit value statements are not in nLab entries.
I added to valuation some mention of the valuation ring, valuation ideal, and valuation uniformity, as well as some examples.
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