Lagmental Vicfred

Formal Power Series Complete a Polynomial Ring at x by Vicfred

Last updated: Fri 08 July 2022

tags: mathematics, completion, filtrations, Rees algebras

The x-adic completion of k[x] is the formal power-series ring k[[x]]. The point is to make the formal expression readable enough to audit line by line.

Statement

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 \(I^n\).

$$ \widehat{k[x]}^{\,(x)}=\varprojlim_nk[x]/(x^n) $$

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.

$$ \widehat{k[x]}^{\,(x)}\cong k[[x]] $$

Worked algebra

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ (a_0,\ a_0+a_1x,\ \ldots,\ \sum_{i=0}^{n-1}a_ix^i,\ldots)\longleftrightarrow\sum_{i\ge0}a_ix^i $$

Conceptual compression

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

$$ \begin{aligned} \mathsf{D}\;&:\quad \widehat{k[x]}^{\,(x)}=\varprojlim_nk[x]/(x^n),\\[5pt] \mathsf{C}\;&:\quad \widehat{k[x]}^{\,(x)}\cong k[[x]]. \end{aligned} $$

Caveat

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \widehat{k[x]}^{\,(x)}\cong k[[x]] \end{gathered}} $$

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