Brauer-Severi schemes of finitely generated algebras
Let \(A\) be a finitely generated algebra over a commutative ring \(R\). The author proves that the Brauer-Severi scheme of \(A\) of a fixed degree \(n\) [see \textit{M. Van den Bergh}, in: Perspectives in Ring Theory, NATO ASI Ser., Ser. C 233, 333-338 (1988; Zbl 0761.13003)] is isomorphic to \(\text{Proj}(Q_{A,n})\) for some graded commutative \(R\)-algebra \(Q_{A,n}\). It is shown that this algebra is generated by a subset of semi-invariants relative to the diagonal \(\text{GL}_n\)-action on the fibered product of the scheme of representations of \(A\) of rank \(n\) with \(\mathbb{A}^n_R\). This result generalizes Corollary~1.11 of [\textit{G. Seelinger}, J. Algebra 184, No. 3, 852-882 (1996; Zbl 0884.16015)].
scientific article; zbMATH DE number 1356482
scientific article