A description of the Brauer-Severi scheme of trace rings of generic matrices
Let \(k\) be an algebraically closed field of characteristic 0, and let \(T\) denote the trace ring of \(m\) generic \(n\times n\) matrices over \(k\). The paper under review gives a description of the Brauer-Severi scheme for \(T\) (i.e., for the fibers of \(T\)). It is shown that the Brauer-Severi scheme for \(T\) is isomorphic to \(\text{Proj}(Q)\) for an appropriate (and described with its generators) graded ring \(Q\). Another important contribution in this paper is the study of the local properties of \(T\) considered as a sheaf over \(\text{Spec}(C)\) where \(C\) is the centre of \(T\). Sufficient conditions are found for the Brauer-Severi scheme of the fiber of \(T\) having smooth irreducible components that meet transversally. In addition, the case \(m=n=2\) is dealt with as an important example. The paper contains new interesting results; it is organized in an appropriate way, and provides sources for further research.
scientific article; zbMATH DE number 941270