The “Defektsatz” for central simple algebras

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.