Excellent algebraic groups. I
Knebusch has defined a regular quadratic form \(q\) over a field \(k\) to be excellent if for any field extension \(K\) of \(k\), there exists a form \(\eta\) over \(k\) such that \(\eta\otimes_kK\) is the anisotropic kernel of the form \(q\otimes_kK\). The authors define a semisimple algebraic \(k\)-group \(G\) to be excellent, if for any field extension \(K\) of \(k\), there exists an algebraic group \(H\) over \(k\) such that \(H\times_kK\) is the anisotropic kernel of the \(K\)-group \(G\times_k K\). In the first part of the paper the authors extend Knebusch's definition of excellence to an Azumaya algebra \(A\) over \(k\) and prove that the following are equivalent: (i) \(A\) is excellent; (ii) the special linear group \(\text{SL}_1(A)\) is excellent; (iii) the index of \(A\) is squarefree. In the second part of the paper the authors consider the special orthogonal group \(\text{SO}(q)\) of a regular quadratic form \(q\) over an infinite field \(k\) of characteristic \(\neq 2\). If \(q\) is excellent it is clear that \(\text{SO}(q)\) is excellent. However the authors prove that excellent special orthogonal groups can arise from many non-excellent quadratic forms. On the other hand, for the special case where the \(k\)-dimension of \(q\) is odd, it follows from their results that \(\text{SO}(q)\) is excellent if and only if \(q\) is excellent.
scientific article; zbMATH DE number 1139781
scientific article