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

Abstract

Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). Then we reprove by a new method that the natural surjective ring map $R\rightarrow k$ admits a splitting if and only if $\Char(R)=\Char(k)$. In our proof there is no need for the existence of the coefficient field for equi-characteristic complete local rings, whose existence is the hardest part of the known proof. However, the main result of this article is that in the unequal characteristic case $\Char(R)\neq\Char(k)$, we prove that the natural surjective map between the groups of units $R^{\ast}\rightarrow k^{\ast}$ admits a splitting. As an application of the above theorem, 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. Next, we show with an example that the above exact sequence does not split for many incomplete local rings.

Theorems & Definitions (56)

• Theorem 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Corollary 1.5
• Corollary 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof