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 E be an O-Y-order in a finite dimensional semisimple algebra A over K = k(Y). Then, for any automorphism tau of A that restricts to an automorphism sigma of Y and any ample, invertible E-bimodule B, Van den Bergh constructs a noetherian, "twisted homogeneous coordinate ring" B = +H-0(Y,B x ... x B-tau n=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 is an element of Y at which sigma has finite order. This allows one to give detailed information about the structure of R and its modules, (C) 2000 Academic Press.