Stability of unitary \(\text{SK}_1\) of Azumaya algebras.
Let \(A\) be an Azumaya algebra with center \(C\) of rank \(n^2\) and a unitary involution \(\tau\). In the paper under review the author studies the stability of the unitary \(\text{SK}_1\) under reduction. The main result is the following theorem. Theorem 1. Let \(A\) be an Azumaya algebra of rank \(n^2\) over its center \(C\) with a unitary involution \(\tau\) and \(R=C^\tau\), where \(R\) is a Henselian ring and \(2\) and \(n\) invertible in \(R\). Then \(\text{SK}_1(A,\tau)\cong\text{SK}_1(\overline A,\overline\tau)\), where \(\overline A=A/mA\) and \(m\) is the unique maximal ideal of \(R\). Recall that the stability of \(\text{SK}_1\) was previously proved by Platonov for an unramified valued division algebra (over a complete discretely valued field) and by Yanchevskii (over a discretely valued Henselian field). Here, using Theorem 1 above, the author gives another proof of the stability of unitary \(\text{SK}_1\) for the general case of valued division algebras over Henselian fields.
valued division algebras over Henselian fields
scientific article; zbMATH DE number 6006284
scientific article