Semiprime Graded Algebras of Dimension Two

Semiprime, noetherian, connected graded k-algebras R of quadratic growth are described in terms of geometric data. A typical example of such a ring is obtained as follows: Let Y be a projective variety of dimension at most one over the base field k and let be an Y-order in a finite dimensional semisimple algebra A over K = k(Y). Then, for any automorphism τ of A that restricts to an automorphism σ of Y and any ample, invertible -bimodule , Van den Bergh constructs a noetherian, “twisted homogeneous coordinate ring” B = H0(Y, ••• τn − 1). We show that R is noetherian if and only if some Veronese ring 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 Y at which σ has finite order. This allows one to give detailed information about the structure of R and its modules.