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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2412.19224 |
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| _version_ | 1866916552051261440 |
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| author | Adachi, Norio |
| author_facet | Adachi, Norio |
| contents | Let $v$ be a discrete valuation of a field $K$, which indicates that the valuation group of $v$ is isomorphic to the integers $\mathbb{Z}$ with the natural order, and let $L$ be a finite separable extension of $K$ with a complete set $\{V_1,V_2,...,V_g\}$ of extended valuations of $v$. Then it is well-known that the following basic equation holds: \[\sum_{j=1}^{g} e_jf_j= [L:K],\] where $e_j$ and $f_j$ denote the ramification index and the relative degree for each $j$, respectively. We extend this result to the case when $v$ is a semi-discrete valuation, indicating that the valuation group is isomorphic to $\mathbb{Z}^n\ (n\geq 1)$ with lexicographic order. As a corollary to this result, we show that it is necessary and sufficient for the integral closure $D$ of the valuation ring $A$ of $v$ to be a free $A$-module that all prime ideals of $D$ other than the maximal ideals are unramified. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19224 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Equality $\sum_{j} e_jf_j=[L:K]$ \vspace{2mm} On the Equality $\sum_{j} e_jf_j=[L:K]$ for a Finite Separable Extension $L$ of $K$ Adachi, Norio Commutative Algebra Let $v$ be a discrete valuation of a field $K$, which indicates that the valuation group of $v$ is isomorphic to the integers $\mathbb{Z}$ with the natural order, and let $L$ be a finite separable extension of $K$ with a complete set $\{V_1,V_2,...,V_g\}$ of extended valuations of $v$. Then it is well-known that the following basic equation holds: \[\sum_{j=1}^{g} e_jf_j= [L:K],\] where $e_j$ and $f_j$ denote the ramification index and the relative degree for each $j$, respectively. We extend this result to the case when $v$ is a semi-discrete valuation, indicating that the valuation group is isomorphic to $\mathbb{Z}^n\ (n\geq 1)$ with lexicographic order. As a corollary to this result, we show that it is necessary and sufficient for the integral closure $D$ of the valuation ring $A$ of $v$ to be a free $A$-module that all prime ideals of $D$ other than the maximal ideals are unramified. |
| title | On the Equality $\sum_{j} e_jf_j=[L:K]$ \vspace{2mm} On the Equality $\sum_{j} e_jf_j=[L:K]$ for a Finite Separable Extension $L$ of $K$ |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2412.19224 |