Splitting in a complete local ring and decomposition its group of units

TL;DR

This paper proves splitting results for the group of units in complete local rings, providing new proofs and applications, including explicit decompositions of the unit group.

Contribution

It introduces novel splitting proofs for unit groups in complete local rings, avoiding complex coefficient field constructions in equi-characteristic cases.

Findings

In unequal characteristic, the map $R^{ imes} o k^{ imes}$ splits.

In equi-characteristic, the map $R o k$ admits a splitting without coefficient fields.

The exact sequence $1 o 1+M o R^{ imes} o k^{ imes} o 1$ always splits for complete local rings.

Abstract

Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). As the first main result of this article, we prove that in the unequal characteristic case $\Char(R)\neq\Char(k)$, the natural surjective map between the groups of units $R^{\ast}\rightarrow k^{\ast}$ admits a splitting. \\ Next, we reprove by a new method that in the equi-characteristic case $\Char(R)=\Char(k)$, the natural surjective ring map $R\rightarrow k$ admits a splitting. In our proof there is no need for the existence of the coefficient fields for equi-characteristic complete local rings, whose existence is the most difficult part of the known proof. \\ As an application, we show that for any complete local ring $(R,M,k)$ the following short exact sequence of Abelian groups: $$\xymatrix{1\ar[r]&1+M\ar[r]& R^{\ast}\ar[r]&k^{\ast} \ar[r]&1}$$ is always split. In particular, we have an isomorphism of Abelian groups $R^{\ast}\simeq(1+M)\times k^{\ast}$. We also show with an example that the above exact sequence does not split for many incomplete local rings.