Lagmental Vicfred

Regular Local Rings Match Dimension with Tangent Dimension by Vicfred

Last updated: Fri 25 December 2015

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

A Noetherian local ring is regular when its maximal ideal needs exactly dim A generators. A small computation will anchor the general statement before the abstraction takes over.

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 \(k=A/\mathfrak m\).

$$ (A,\mathfrak m,k)\ \text{Noetherian local} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ A\ \text{regular}\Longleftrightarrow\dim A=\dim_k\mathfrak m/\mathfrak m^2 $$

Push the symbols

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ A=k[x_1,\ldots,x_n]_{(x_1,\ldots,x_n)},\qquad\mathfrak m/\mathfrak m^2=\bigoplus_{i=1}^{n}k\bar x_i,\qquad\dim A=n $$

Structural reading

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad (A,\mathfrak m,k)\ \text{Noetherian local},\\[5pt] \mathsf{C}\;&:\quad A\ \text{regular}\Longleftrightarrow\dim A=\dim_k\mathfrak m/\mathfrak m^2. \end{aligned} $$

A hypothesis worth keeping

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] A\ \text{regular}\Longleftrightarrow\dim A=\dim_k\mathfrak m/\mathfrak m^2 \end{gathered}} $$

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 Thu 17 September 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.