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created absolute value and a stubby Ostrowski’s theorem
Cf. related entries valuation and discrete valuation.
Yes, I saw them. Let’s discuss a bit what’s going on here. What’s called a valuation at valuation reduces just to non-archimedean valuations for $G = \mathbb{R}$. I know that “valuation” and “absolute value” are used inconsistently in the literature, but what the entry valuation currently describes seems to add a new layer of inconsistency.
Am I wrong? Otherwise we should fix that.
Well, you have two versions of valuations, the one is exponential of another. The exponential version is sometimes called also the multiplicative valuation. The entry valuation is very standard, once this is taken into account.
We should add more explanation and disambiguation. Also valuation of measure spaces needs to be discussed or at least pointed to at “valuation”.
One really says valuation of measure space ? I am surprised a bit, range of measure or alike would be better.
From valuation:
Sometimes one also discusses exponential (or multiplicative) valuations (also called valuation functions, and viewed as generalized absolute values) which look more like norms, and their equivalence classes, places. See discrete valuation and valuation ring.
One really says valuation of measure space ?
Then I think it should definitely be a separate entry like on wikipedia.
An important idea (from Dedekind and Weber) about the geometry coming from valuations is sketched in the entry Riemann surface via valuations.
Sure, we should make disambiguations.
I’m confused by this:
The standard absolute value on the complex numbers is the absolute value of the real part.
Surely the standard absolute value of $a + b\mathrm{i}$ is $\sqrt{a^2 + b^2}$, not ${|a|} = \sqrt{a^2}$? This satisfies the axioms on the page and also extends (if you allow $k$ to be any ring) to the standard absolute value on the quaternions or even the octonions. I’ve never heard of this other absolute value, and I’m not sure how it (or any absolute value) is supposed to make the complex numbers into an archimedean field (or indeed any ordered field, as proved impossible at ordered field).
Woops, did I type that? (Looks like I did :-/). I didn’t mean to. I have fixed it now. Thanks for catching this.
Do you still want to say that $\mathbb{C}$ is an archimedean field? You say this also at archimedean field, so you presumably mean something by it. Is it just that the absolute value is always finite? But the meaning of archimedean field that I know (and on that page) is a property of an ordered field.
What I want to say is that $\mathbb{C}$ is an archimedean valued field, i.e. complete with respect to an archimedean valuation.
I’ll try to clarify the entries…
I have tried to correct/clarify, in the course of which I also created an entry archimedean valued field. There is still room for more clarification and more details, of course.
But this ambiguous use of terminology seems to be not just my fault, but be in the standard literature. On page 12 of
we have
A non-Archimedean field is a valuation field with non-Archimedean valuation.
and then
If a valuation field is not non-Archimedean, it is isomorphic to $\mathbb{R}$ or to $\mathbb{C}$
(okay, I recorded that at Ostrowski’s theorem) but then
Such fields are said to be Archimedean.
Another thing that we still need to clarify more in the $n$Lab is that what Berkovich calls a “valuation” is not what our entry “valuation” defines, but what our entry absolute value defines. Whatever we settle for, eventually there need to be more alerts about different use of terminology.
I put a terminology warning at absolute value.
Another thing that we still need to clarify more in the nLab is that what Berkovich calls a “valuation” is not what our entry “valuation” defines, but what our entry absolute value defines.
Urs, I am repeating again, what I sadi above, and this IS contained in the circle of valuation entries (including discrete valuation) that there are MULTIPLICATIVE valuations and ordinary valuations and that absolute values are a SPECIAL CASE of multiplicative valuations, so that multiplicative valuations are thought of as generalized absolute values. This is a standard terminology and we should not alienate experts in those fields by introducing new confusion. By exponentiating ordinary valuations we get multiplicative valuations (therefore also called exponential valuations). It is not only Berkovich school but large part of mathematics which uses this standard terminology. Multiplicative valuations differ from other kind just by exponentiation. It would not be good to rename multiplicative valuations as qabsoluet values as the level of generality and terminology here have standard values in large portion of mathematics including arithmetic and part of algebra and algebraic geometry. Important thing is also the equivalence of valuations.
Look at entry discrete valuation.
P.S. maybe it is a time to look at valuative criterion of properness.
No, Zoran, what valuation currently defines is not what Berkovich (nor Wikipedia, for that matter) calls a valuation and is not equivalent to it. For one, in its last clause is the restriction to the non-Archimedean case. More generally there should be the triangle inequality there.
And concerning alienation: I would like to reduce any possible alienation by pointing out different uses of terminology in the literature.
Erased.
No, Zoran, what valuation currently defines is not what Berkovich (nor Wikipedia, for that matter) calls a valuation
It seems to me that Wikipedia agrees with what we currently have at valuation. I haven’t looked at Berkovich.
You are right about Wikipedia. My fault. Sorry for saying the opposite. Not sure why I thought it would be different there.
Let me list some sources:
Berkovich in all his works (for instance in the first definfinition of his his lectures or on the first technical page of his book) writes “valuation” for the thing that is multiplicative and satisfies the triangle identity. He calls it “non-archimedean valuation” if the triangle identity is refined to the ultrametric property.
This is also the convention used by Wolfram MathWorld, see valuation (Wolfram MathWorld).
The same for PlanetMath, see valuation (PlanetMath).
Urs, do you have access to the famous collection of lectures on number theory edited by Froehlich ? It has several authors which have a different point of view on generality, but do make comparison remarks on general conventions. It is the main source on the subject.
Interesting sister variant – a weight on a von Neumann algebra – see e.g. Lurie lec. 34 pdf, def. 3.
the definition-section at absolute value first says that it’s a multiplicative semi-norm, but then it spells out the axioms of a multiplicative norm. We should fix this either way, it seems.
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