Valuation Rings

I want to talk a bit about valuation rings. As a general topic, valuation rings are rather new to me, so hopefully my explanations go smoothly. The only prior knowledge I will assume is that of local rings and some elementary facts about integral elements.

To begin, let K be a field. Then a valuation on K is a map \nu:K^\times\longrightarrow\mathbb R such that the following hold for all a,b\in K^\times:

  1. \nu(ab)=\nu(a)+\nu(b),
  2. \nu(a+b)\geq\min\{\nu(a),\nu(b)\}.

Notice that the first condition means that \nu is a group homomorphism from the multiplicative group K^\times to the additive group \mathbb R, hence the image \Gamma=\nu(K^\times) is a subgroup of the real numbers, which we call the valuation group of \nu. We will set \nu(0)=\infty, which is assumed to be larger than any real number. Now any such pair (K,\nu) of a field K and valuation \nu on K is called a valuation field.

Now let (K,\nu) be any valuation field. We then easily see the following properties:

  1. \nu(\pm1)=0,
  2. \nu(-a)=\nu(a) for all a\in K,
  3. \nu(a^{-1})=-\nu(a) for all a\in K^\times.

If we define R:=\{x\in K:\nu(x)\geq0\}, we see that R is a subring of K called the valuation ring of \nu. Since R is a subring of a field, it is an integral domain. If we set R^{-1}:=\{x^{-1}\in K:0\neq x\in R\}, we see that K=R\cup R^{-1} and K is the field of fractions of R.

Notice that \mathfrak m:=\{x\in R:\nu(x)>0\} is an ideal of R, called the valuation ideal of \nu. Now suppose that x is a unit of R. Then x,x^{-1}\in R, implies \nu(x),\nu(x^{-1})\geq0. But \nu(x^{-1})=-\nu(x), implying that \nu(x)=0. Therefore R^\times=\{x\in R:\nu(x)=0\}. Since the set of non-units \mathfrak m is an ideal of R, we see that R is a local ring. By property of local rings, \mathfrak m is maximal and k:=R/\mathfrak m is a field, called the residue field of \nu.

We want to show one more interesting property of valuation fields before moving forward. Suppose \nu(b)<\nu(a) for some a,b\in K. We then see that \nu(a)-\nu(b)=\nu(\frac ab)>0, implying \frac ab\in\mathfrak m. Then 1+\frac ab must be a unit of R since otherwise 1+\frac ab\in\mathfrak m would imply that 1+\frac ab-\frac ab=1\in\mathfrak m, an obvious contradiction. From this we have

0=\nu(1+\frac ab)=\nu(\frac{a+b}b)=\nu(a+b)-\nu(b).

So we have proven that \nu(b)<\nu(a) implies \nu(a+b)=\nu(b).

Next we want to discuss the idea of equivalent valuations on K. For two valuations \nu,\nu' on K, we say \nu and \nu' are equivalent and write \nu\sim\nu' if there exists a positive real number \lambda such that \nu'(a)=\lambda\nu(a) for all a\in K. This then defines an equivalence relation on valuations on K. It is easy to see that equivalent valuations given the same valuation ring, valuation ideal, and group of units. A valuation is called discrete if its value group is an infinite cyclic group, i.e., \nu(K^\times)\cong\mathbb Z. If the value group is equal to – not simply isomorphic to – the integers, then the valuations is called a normalized discrete valuation. Notice that every discrete valuation is equivalent to a normalized discrete valuation.

We want to focus for a bit on discrete valuations rings, or simply DVRs. So let \nu be a normalized discrete valuation on on a field K with valuation ring R and valuation ideal \mathfrak m. Then the following properties hold:

  1. Let \pi\in R be an element such that \nu(\pi)=1. Then any \alpha\in K can be expressed as \alpha=u\pi^i where i=\nu(\alpha) and u is a unit of R.
  2. Any nonzero ideal of R is of the form \mathfrak m^i=(\pi^i) for some i\geq0. In particular, R is a PID.
  3. R is integrally closed.

Such an element \pi in the above is called a uniformizing parameter of R. From here we can see that two discrete valuations on a fixed field are equivalent if and only if they have the same valuation ideal. We finish this section with a result that gives a characterization of DVRs. It can be shown that the following two conditions on an arbitrary ring R are equivalent:

  1. R is a DVR.
  2. R is a local PID.

Next time we discuss \mathfrak p-adic valuations and complete valuation rings.

Summer Reading List

I’m posting all the books I plan on reading in the next couple of months, listed in order from least academic to most academic. Anyone reading this should feel free to add reading suggestions or leave comments if you’ve read them also.

  • “Hitchhiker’s Guide to the Galaxy” – Douglas Adams
  • “The Girl with the Dragon Tattoo” – Stieg Larsson
  • “Surely You’re Joking, Mr. Feynman!” – Richard Feynman
  • “The Psycopath Test” – Jon Ronson
  • “Letters to a Young Mathematician” – Ian Stewart
  • “Stiff: The Curious Lives of Human Cadavers” – Mary Roach
  • “The Believing Brain” – Michael Shermer
  • “The Drunkard’s Walk: How Randomness Rules Our Lives” – Leonard Mlodinow
  • “Thinking, Fast and Slow” – Daniel Kahneman
  • “Atheism, Atheology, and Secular Philosophy” – John Shook
  • “Representations of Finite Groups” – Hiroshi Nagao, Yukio Tsushima

An explanation

Though I don’t think starting my own blog necessitates an explanation, I feel compelled to explain myself and my goals for this blog. So here it goes…

When I was younger, I saw myself as being highly capable in both mathematics and writing. While I have proven my mathematical ability to a high enough degree to warrant getting into a PhD program (where I’m still going strong!), somewhere along the wa, I seem to have let my writing ability fall behind. I can think of a couple of possible explanations for this, both stemming from the same root problem. First of all, I do continue to write quite a bit. The problem is however, that almost all of my writing is mathematical writing. The problem here is that very little of this writing is creative or even involves words; mostly it is proof writing involving abstract symbolism. A separate – though intimately related problem – is that I have noticed this issue in my writing ability and shrugged it off with the assumption that it is unimportant since my future seems to be headed towards math research and education. This thinking is flawed for several reasons. First of all, given the competition involved in the job market, it is possible I may never become a research mathematician (a horrible but realistic possibility). Secondly, there is no reason to assume a mathematician could not – or should not – be a proficient writer.

So what am I doing here? Well, as explained above, it’s obvious I need practice writing since practice is the surely the best way to improve. I see a blog as a constructive way to practice as well as allowing possible critiques and criticisms from anyone I might show this blog to or anyone that might stumble in on their own volition. This seems preferred over a personal journal since it could be otherwise hard to gauge improvements and the idea that other people can read what I write down will hopefully force me to put greater effort into expressing my thoughts. I must admit that this is my foremost reason for this blog. At times I find myself having very detailed and thought-out opinions about various issues. It is in expressing my opinions in words that I often fall flat. This will be an exercise in becoming better at expressing my thoughts and opinions through words. Hopefully this will prove to be an effective strategy.

Lastly, I’d like to discuss briefly what I intend to include for general content of this blog. I am foremost a mathematics student so it’s very probable some of my work will appear here, though likely in a non-rigorous fashion. The appeal of mathematics to me is both in its abstraction and its definitiveness. There is room for creative thinking, but in most situations a proposition can explicitly be shown to be true or false. Outside of mathematics this becomes increasingly difficult, but logic and mathematical thinking can be applied quite often in everyday life. I am a self-proclaimed skeptic and hope to use this page to turn a skeptical eye on issues of the day that interest me. Rationality is a tool I use frequently in attempts to better myself, and I hope to explain what this might look like in ordinary situations we encounter every day.

First Post

Hello Internet World. This is not my first attempt at a blog. However, I am often highly critical of my own work and have previously (I think) purged the Internet of anything resembling a blog from me. There have been many times where I have contemplated starting up a blog again, and I think I now have enough projects going on that it is almost become a necessity for organizing my own thoughts. More info on what I plan to do with this blog will be posted once I get some things in order and get a feel for this site.

 

Enjoy,

Rob