On the algebra of some group schemes. (Q1005873)

Let \(k\) be a semilocal ring containing \(\mathbb{Q}\), and let \(A\) be a projective separable (not necessarily commutative) \(k\)-algebra. In the work under review, the author proves the existence of a finite étale \(k\)-group scheme \(G\) and a surjective \(k\)-algebra map \(k\langle G\rangle\to A\), i.e. \(A\) is the quotient of the algebra of a finite étale group scheme. The group \(G\) and the surjection are constructed as follows. Let \(K\) be the center of \(A\), and let \(L\) be an étale maximal subalgebra of \(A\). One can then define a sequence \(1\to R_{L/k}(\mu_{n,L})\to G_0\to\Aut(L/K)\to 1\) of sheaves in the étale topology, where \(R_{L/k}\) is the Weil restriction functor. Then the desired group \(G\) is obtained by a pull-back of \(G_0\) using the sequence \(1\to R_{L/k}(\mu_{n,L})\to G\to W\to 1\), where \(W\) is a \(k\)-subgroup of \(\Aut(L/K)\) with \(L^W=K\). The map \(f\colon k\langle G\rangle\to A\) is the map associated to the inclusion \(G\hookrightarrow\mathbf G_{m,A/k}\). Some specific examples are given, including showing how a matrix algebra can be generated from finite groups, as well as how the quaternions can be. The reader will find this paper thoroughly written with very clear explanations. It is presented as an analogy to the Brauer-Witt classification of projective separable algebras over an algebraically closed field which arise as the algebra of a finite group, however (from the paper): ``[d]espite a formal analogy with the Brauer-Witt theory, our result does not add much to it: even in the simplest case, that of the quaternions, our method gives a \textit{nonconstant} group scheme for generating this \(\mathbb{R}\)-algebra, in fact a group which is a definitely twisted form of the dihedral group \(D_4\)''.