Prescribed behavior of central simple algebras after scalar extension. (Q427766)

Let \(A_1,\dots,A_n\) be disjoint central simple algebras over a field \(F\), for some positive integer \(n\), and for each central simple \(F\)-algebra \(B\), let \(\text{ind}(B)\) be the Schur index and \(\exp(B)\) the exponent of \(B\), respectively. Denote by \(B^m\) the \(m\)-th tensor power of \(B\) over \(F\) whenever \(m\in\mathbb N\), and put \(B^{-m}=B^{-1(m)}\), where \(B^{-1}\) stands for the (central simple) \(F\)-algebra \(B^{\text{op}}\) inversely-isomorphic to \(B\). We say that \(A_1,\dots,A_n\) are disjoint, if NEWLINE\[NEWLINE\text{ind}(A_1^{j(1)}\otimes_F\cdots\otimes_FA_n^{j(n)})=\text{ind}(A_1^{j(1)})\cdots\text{ind}(A_n^{j(n)}),NEWLINE\]NEWLINE for all integers \(j(1),\dots,j(n)\). Assuming that \(A_1,\dots,A_n\) are disjoint, fix positive integers \(l_1,m_1,\dots,l_n,m_n\), such that for each index \(i\), \(l_i\) divides \(\exp(A_i)\) and \(m_i\), \(m_i\mid\text{ind}(A_i)\), and \(l_i\) and \(m_i\) have the same prime divisors.NEWLINENEWLINE The paper under review shows that then there exists a finitely-generated regular field extension \(E/F\), such that \(\exp(A_i\otimes_FE)=l_i\) and \(\text{ind}(A_i\otimes_FE)=m_i\), \(i=1,\dots,n\).NEWLINENEWLINE This continues the study of the behaviour of Schur indices and exponents of central simple \(F\)-algebras under scalar extensions of \(F\), carried out in an earlier paper by the authors, [see ``Symbol algebras and cyclicity of algebras after a scalar extension'', Fundam. Prikl. Mat. 14, No. 6, 193-209 (2008); translation in J. Math. Sci., New York 164, No. 1, 131-142 (2010)]. Using ideas similar to those in their earlier paper, they prove that if \(K/F\) is a quadratic field extension and \(A\) is a central simple algebra over \(K\) with an involution \(\tau \) of the second kind trivial on \(F\), then there exists a regular field extension \(E/K\) preserving Schur indices of central simple \(K\)-algebras, such that \(A\otimes_KE\) is cyclic and has involution of the second kind extending \(\tau\). As an application, the authors prove reduction theorems on a unitary variant of Suslin's conjecture about the \(R\)-nontriviality [\textit{V. Chernousov} and \textit{A. J. Merkurjev}, Algebra 209, No. 1, 175-198 (1998; Zbl 0947.14023)] of the special unitary group \(\text{SU}(A,\tau)\). Specifically, they show that it is sufficient to prove this variant for cyclic division algebras of degree \(p^2\), which are presentable as tensor products of two cyclic algebras of degree \(p\).