Lagmental Vicfred

Nakayama's Lemma Removes Redundant Local Generators by Vicfred

Last updated: Fri 13 November 2020

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

A finite module over a local ring vanishes when multiplication by the maximal ideal already fills it. I want the notation, the mechanism, and the failure mode visible at the same time.

Objects and notation

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

$$ (A,\mathfrak m)\ \text{local},\qquad M\ \text{finite} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \mathfrak mM=M\Longrightarrow M=0 $$

Push the symbols

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ M/\mathfrak mM=0\quad\Longrightarrow\quad M=\mathfrak mM\quad\overset{\text{Nakayama}}{\Longrightarrow}\quad M=0 $$

Structural reading

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad (A,\mathfrak m)\ \text{local},\qquad M\ \text{finite},\\[5pt] \mathsf{C}\;&:\quad \mathfrak mM=M\Longrightarrow M=0. \end{aligned} $$

A hypothesis worth keeping

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathfrak mM=M\Longrightarrow M=0 \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Tue 23 June 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.