Learn about valuations, valuation rings, value group

Question

I am reading a paper for a summer research project (Example of an interpolation domain ). I am unfamiliar with some of the terms used here and I have tried searching on google for definitions but I am a little confused. I am not sure if the definitions I am finding pertain to what I am looking at. The passage that I am at states:

... we take $v(r)$ to be the smallest exponent on a nonzero term of $r$. Denote the valuation ring of $v$ by $V$; so that, to be in $V$, a series must have no (nonzero) terms with negative exponents. Because $V$ has value group $G$, its maximal ideal is idempotent, and hence ...

(The quote comes from the bottom of the first paragraph in Example 1 on page 2.)

We are doing long division of polynomials and writing them as a kind of Laurent series and each non-zero element of the fraction field has a unique kind of "Laurent expansion." I have worked a little bit with valuations before but in what seemed to be kind of informal as the definition that I worked with then I have not seen repeated in my searches recently. Specifically, I worked with the $p$-adic valuation on $\mathbb{Z}$ defined by $v_p(x)=p^{-n}$ where $x=p^nb$, $p\nmid b$ for $x\neq 0$ and $v_p(x)=0$ if $x=0$.

I just want to learn more about valuations, valuation rings, value groups, and an ideal being idempotent. From what I gather, you define a valuation on a field and then this gives rise to a valuation ring. The definitions of valuation that I have looked at involve $\infty$ which confuses me a little bit as it looks like each non-zero element gets mapped to a rational number in this case, as the exponents are rational. Looking at when I worked with the $p$-adic valuation on $\mathbb{Z}$, we never mapped $0 \mapsto \infty$.

Maybe there is material that is available online I can look at to work through elementary results with valuations and valuation rings. Any advice on places I can go to find more information about these things? Thanks very much.

Answer 1

There are several places to learn valuation theory.

• A good place is, our very own, Pete L Clark's notes: http://alpha.math.uga.edu/~pete/8410Chapter1.pdf (more advanced topics by Pete [on the same subject] can be found here: http://alpha.math.uga.edu/~pete/MATH8410.html).

• My personal favorite though, albeit somewhat strange would have to be the beginning to Local Class Field Theory by Iwasawa (just the first chapter!). It may be hard to get your hands on this though, as it is out of print. There are, ahem, websites to find such materials though.

There are many places NOT to look as a first go (in my opinion), especially considering your apparent goals. Among these is Serre's Local Fields.

If you get really desperate (and I mean really), I have a set of notes I wrote that were a conglomeration of, and sometimes reworded/with my commentary mixed in, notes on valuation theory. If you'd like I could send them to you--but this is a last resort.

Answer 2

I think a good place to start would be not to learn about general valuations but more specific discrete valuations and discrete valuation rings. For any field $K$, a discrete valuation on $K$ is a function $\nu$ from $K^\ast \to \Bbb{Z} \cup \{\infty\}$ that satisfies

1. $\nu(xy) = \nu(x) + \nu(y)$
2. $\nu(x+y) \geq \min (\nu(x),\nu(y))$.

In this context, the valuation ring can be defined to be the set of all $x \in K$ such that $\nu(x) \geq 0$. For example we see immediately that the valuation ring of the $p$ - adic valuation (which is a discrete valuation on $\Bbb{Q}$) is the localization $\Bbb{Z}_{(p)}$. One other motivation for knowing about discrete valuation rings is that the local ring at any point of an algebraic number field is always a DVR (more generally the local ring at any point of a Dedekind domain is always a DVR). Knowing these facts is very useful in proving unique factorization of ideals, that the quotient of any Dedekind domain is always a PIR, etc.

For a reference on discrete valuation rings, I suggest Miles Reid's Undergraduate Commutative Algebra. The book is quite readable, but if I remember IIRC to read the chapter on DVR's you need to know basic commutative algebra. If you're adventurous, you can look at Atiyah - Macdonald and try the exercises at the end of chapter 5 on general valuation rings, or for DVRs chapter 9.