A class of noncommutative projective surfaces

Rogalski, D. and Stafford, J T (2009) A class of noncommutative projective surfaces. Proc London Math Soc, 99. pp. 100-144. ISSN 1749-9097

PDF ptschemes-journal.pdf Download (586kB)

Abstract

Let A=k+A_1+A_2.... be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A)=k(Y)[t,t^{-1},sigma], where sigma is an automorphism of the integral projective surface Y. Then we prove that A can be written as a naive blowup algebra of a projective surface X birational to Y. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in the quotient category qgr A will always be in (1-1) correspondence with the closed points of the scheme X.

Item Type:	Article
Uncontrolled Keywords:	{Noncommutative projective geometry, noncommutative surfaces, noetherian graded rings, naive blowing~up
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 14 Algebraic geometry MSC 2010, the AMS's Mathematics Subject Classification > 16 Associative rings and algebras
Depositing User:	Professor J T Stafford
Date Deposited:	15 May 2013
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1975

Available Versions of this Item

• A class of noncommutative projective surfaces. (deposited 26 Nov 2007)
  • A class of noncommutative projective surfaces. (deposited 15 May 2013) [Currently Displayed]

Actions (login required)

View Item