Semiprime graded algebras of dimension two
In a previous article [Invent. Math. 122, No. 2, 231-276 (1995; Zbl 0849.16022)], the authors have classified Noetherian graded domains of quadratic growth over algebraically closed fields; the paper under review extends this classification to semiprime algebras over an arbitrary base field \(k\). In some sense, the description of noncommutative projective curves that was begun in the earlier paper can thus be considered complete. Let \(R\) be a semiprime, Noetherian, connected graded \(k\)-algebra of quadratic growth. The description of \(R\) in terms of geometric data proceeds, roughly speaking, along the following lines: Let \(Y\) be a projective variety of dimension \(\leq 1\) over the base field \(k\) and let \(\mathcal E\) be an \({\mathcal O}_Y\)-order in a finite dimensional semisimple algebra \(A\) over the field \(K=k(Y)\). Also, let \(\tau\) be an automorphism of \(A\) that restricts to an automorphism \(\sigma\) of \(Y\), and let \(\mathcal B\) be an ample, invertible \(\mathcal E\)-module. Following \textit{M.~Van den Bergh} [J. Algebra 184, No. 2, 435-490 (1996; Zbl 0876.17011)], a Noetherian ``twisted homogeneous coordinate ring'' can be constructed as \(B=\bigoplus_n H^0(Y,{\mathcal B}\otimes\cdots\otimes{\mathcal B}^{\tau^{n-1}})\). The ring \(R\) is then Noetherian if and only if some Veronese subring \(R^{(m)}\) of \(R\) has the form \(k+I\), where \(I\) is a left ideal of such a ring \(B\) and where \(I=B\) at each point \(p\in Y\) at which \(\sigma\) has finite order. A lot of detailed information concerning semiprime graded algebras of Gelfand-Kirillov dimension 2 is obtained from this characterization. For example, a finitely generated, semiprime graded Goldie ring \(S\) with \(\text{GKdim}(S)=2\) is non-Noetherian if and only if there exists a second layer link from a prime ideal \(P\) with \(\text{GKdim}(S/P)=1\) to a maximal ideal \(M\) of \(S\).
scientific article; zbMATH DE number 1448638
scientific article