A note on freeness

TL;DR

This paper investigates the conditions under which the power series ring extension is free over its base ring, establishing finiteness of the field extension as a key criterion, and explores the projective dimension of the extension.

Contribution

It precisely characterizes when the power series ring extension is free over the base ring in terms of the finiteness of the field extension.

Findings

Freeness of the extension occurs if and only if the field extension is finite.

Raises the open question about the projective dimension of the power series extension.

Provides a clear criterion linking algebraic properties of field extensions to module theory.

Abstract

In this brief note we show that for a field extension $K/F$, $S=K[\![\mathbf{x}]\!]$ is a free $R=F[\![\mathbf{x}]\!]$-module precisely when $K/F$ is finite. We then raise the question \emph{what is the projective dimension of $S$?}