Discrete valuation ring

A discrete valuation ring (DVR) is the basic local building block of dimension-one commutative algebra.

Definition

A DVR is a local ring $(R,\mathfrak m)$ (see local ring and its maximal ideal ) such that:

• $R$ is a domain,
• $R$ is Noetherian ,
• the maximal ideal $\mathfrak m$ is principal: $\mathfrak m=(\pi)$ for some $\pi\in \mathfrak m$,
• and $R$ has Krull dimension $1$.

Any generator $\pi$ of $\mathfrak m$ is called a uniformizer.

Equivalent characterizations

For a local domain $(R,\mathfrak m)$ that is not a field, the following are equivalent:

1. $R$ is a DVR.
2. $R$ is a principal ideal domain and has a unique nonzero prime ideal (namely $\mathfrak m$).
3. There exists $\pi\in \mathfrak m$ such that every nonzero ideal of $R$ is of the form $(\pi^n)$ for a unique $n\ge 0$.

In particular, in a DVR the ideals form a totally ordered chain:

DVRs arise naturally from Dedekind domains : if $R$ is Dedekind and $\mathfrak p\neq (0)$ is prime, then $R_\mathfrak p$ is a DVR (via localization at $\mathfrak p$ ).

Examples

1. Formal power series.
   If $k$ is a field , then $k[[t]]$ is a DVR with maximal ideal $(t)$ and uniformizer $t$.

2. Localization of a PID at a prime.
   $\mathbb{Z}_{(p)}$ is a DVR: it is local with maximal ideal generated by $p$, and every ideal is $(p^n)$.

3. A polynomial example.
   $k[t]_{(t)}$ (localization of $k[t]$ at the prime $(t)$) is a DVR with uniformizer $t$.

A useful structural fact

In a DVR, the valuation “order of vanishing” is encoded by powers of $\pi$: for $0\ne x\in \mathrm{Frac}(R)$, one can write $x=\pi^n u$ with $u\in R^\times$ and $n\in\mathbb{Z}$, uniquely.