A class of noncommutative projective surfaces

Rogalski, D. and Stafford, J T (2007) A class of noncommutative projective surfaces. [MIMS Preprint]

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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: MIMS Preprint
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: 26 Nov 2007
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/963

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