Lagmental Vicfred

Completion of Finite Modules Is Tensoring with the Completed Ring by Vicfred

tags: mathematics, completion, filtrations, Rees algebras

Over a Noetherian ring, the completion of a finite module is obtained by base change. A small computation will anchor the general statement before the abstraction takes over.

Definitions first

The \(I\)-adic filtration \(A\supset I\supset I^2\supset\cdots\) records increasing orders of vanishing. Completion replaces \(A\) by compatible residues modulo every \(I^n\).

$$ \widehat M^{\,I}=\varprojlim_nM/I^nM $$

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.

$$ M\ \text{finite}\Longrightarrow\widehat M^{\,I}\cong M\otimes_A\widehat A^{\,I} $$

A small case in full

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \begin{aligned}\widehat{A^r/N}^{\,I}&\cong\widehat A^{\,r}/\widehat N,\\0\to\widehat N\to\widehat A^{\,r}\to\widehat M\to0&\quad\text{is exact}.\end{aligned} $$

The reusable statement

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad \widehat M^{\,I}=\varprojlim_nM/I^nM,\\[5pt] \mathsf{C}\;&:\quad M\ \text{finite}\Longrightarrow\widehat M^{\,I}\cong M\otimes_A\widehat A^{\,I}. \end{aligned} $$

A nearby false statement

Completion and localisation answer different questions and do not commute without hypotheses. Completeness is topological data, not merely another quotient.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] M\ \text{finite}\Longrightarrow\widehat M^{\,I}\cong M\otimes_A\widehat A^{\,I} \end{gathered}} $$

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 Tue 15 April 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.