Lagmental Vicfred

A Discrete Valuation Ring Has One Uniformizer by Vicfred

Last updated: Sun 01 May 2022

tags: mathematics, local algebra, Nakayama lemma, regular local rings

Every nonzero ideal of a DVR is a power of its unique maximal ideal. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Set-up

In a local ring \((A,\mathfrak m)\), reduction modulo \(\mathfrak m\) turns finite-module questions into linear algebra over the residue field \(k=A/\mathfrak m\).

$$ (A,\mathfrak m)\ \text{DVR},\qquad\mathfrak m=(\pi) $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \forall\,0\ne I\triangleleft A,\quad I=(\pi^n)\ \text{for a unique }n\ge0 $$

The calculation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \begin{aligned}x&=u\pi^{v(x)},\quad u\in A^\times,\\(x,y)&=(\pi^{\min\{v(x),v(y)\}}).\end{aligned} $$

What survives abstraction

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad (A,\mathfrak m)\ \text{DVR},\qquad\mathfrak m=(\pi),\\[5pt] \mathsf{C}\;&:\quad \forall\,0\ne I\triangleleft A,\quad I=(\pi^n)\ \text{for a unique }n\ge0. \end{aligned} $$

The boundary

Finite generation is essential in Nakayama's lemma. Infinite modules can satisfy \(\mathfrak mM=M\) without vanishing.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \forall\,0\ne I\triangleleft A,\quad I=(\pi^n)\ \text{for a unique }n\ge0 \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Sat 19 June 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.