📝 SummarXiv

Abstract

Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). Then we prove that the natural surjective ring map $R\rightarrow k$ admits a splitting if and only if $\Char(R)=\Char(k)$. In addition, if $\Char(R)\neq\Char(k)$, then we prove that the natural surjective map between the groups of units $R^{\ast}\rightarrow k^{\ast}$ admits a splitting. \\ We also show with an example that the above theorem is in its full generality. As an application of this theorem, we show that 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.