Splitting in a complete local ring and decomposition its group of units
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915939812900864 |
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| author | Tarizadeh, Abolfazl |
| author_facet | Tarizadeh, Abolfazl |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splitting in a complete local ring and decomposition its group of units Tarizadeh, Abolfazl Commutative Algebra Algebraic Geometry Group Theory Rings and Algebras 13B35, 13J10, 12F20, 12F10, 13A35, 13B40, 13B10, 13A15, 13M05, 13D03 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. |
| title | Splitting in a complete local ring and decomposition its group of units |
| topic | Commutative Algebra Algebraic Geometry Group Theory Rings and Algebras 13B35, 13J10, 12F20, 12F10, 13A35, 13B40, 13B10, 13A15, 13M05, 13D03 |
| url | https://arxiv.org/abs/2508.08753 |