Semiprime Graded Algebras of Dimension Two

✍ Scribed by M Artin; J.T Stafford

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 390 KB
• Volume: 227
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8226

✦ Synopsis

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"

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.