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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The ``Defektsatz'' for central simple algebras

Author: Joachim Gräter
Journal: Trans. Amer. Math. Soc. 330 (1992), 823-843
MSC: Primary 16H05; Secondary 16D30, 16K40, 16S70, 16W60
MathSciNet review: 1034663
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Abstract: Let $ Q$ be a central simple algebra finite-dimensional over its center $ F$ and let $ V$ be a valuation ring of $ F$. Then $ V$ has an extension to $ Q$, i.e., there exists a Dubrovin valuation ring $ B$ of $ Q$ satisfying $ V= F \cap B$. Generally, the number of extensions of $ V$ to $ Q$ is not finite and therefore the so-called intersection property of Dubrovin valuation rings $ {B_1}, \ldots,{B_n}$ is introduced. This property is defined in terms of the prime ideals and the valuation overrings of the intersection $ {B_1} \cap \cdots \; \cap {B_n}$. It is shown that there exists a uniquely determined natural number $ n$ depending only on $ V$ and having the following property: If $ {B_1}, \ldots,{B_k}$ are extensions of $ V$ having the intersection property then $ k \leq n$ and $ k= n$ holds if and only if $ {B_1} \cap \cdots \cap {B_k}$ is integral over $ V$. Let $ n$ be the extension number of $ V$ to $ Q$. There exist extensions $ {B_1}, \cdots,{B_n}$ of $ V$ having the intersection property and if $ {R_1}, \ldots,{R_n}$ are also extensions of $ V$ having the intersection property then $ {B_1} \cap \cdots \cap {B_n}$ and $ {R_1} \cap \cdots \cap {R_n}$ are conjugate. The main result regarding the extension number is the Defektsatz: $ [Q:F]= {f_B}(Q/F){e_B}(Q/F){n^2}{p^d}$, where $ {f_B}(Q/F)$ is the residue degree, $ {e_B}(Q/F)$ the ramification index, $ n$ the extension number, $ p = \operatorname{char}(V/J(V))$, and $ d$ a natural number.

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Keywords: Valuation rings, central simple algebras
Article copyright: © Copyright 1992 American Mathematical Society

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