Lagmental Vicfred

Krull's Intersection Theorem Separates I-Adic Orders by Vicfred

Last updated: Fri 06 April 2012

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

In a Noetherian local ring, an element lying in every power of a proper ideal must vanish. I want the notation, the mechanism, and the failure mode visible at the same time.

Notation

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

$$ (A,\mathfrak m)\ \text{Noetherian local},\qquad I\subseteq\mathfrak m $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \bigcap_{n\ge1}I^n=(0) $$

Stress the formula

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

$$ A=k[[t]],\qquad (t)\supset(t^2)\supset\cdots,\qquad\bigcap_{n\ge1}(t^n)=\{0\} $$

Interpretation

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad (A,\mathfrak m)\ \text{Noetherian local},\qquad I\subseteq\mathfrak m,\\[5pt] \mathsf{C}\;&:\quad \bigcap_{n\ge1}I^n=(0). \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \bigcap_{n\ge1}I^n=(0) \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Sun 10 July 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.