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Some projective surfaces of GK-dimension 4

Part of: Chain conditions, growth conditions, and other forms of finiteness Rings and algebras arising under various constructions Rings and algebras with additional structure

Published online by Cambridge University Press:  15 May 2012

D. Rogalski  and
Susan J. Sierra
D. Rogalski
Affiliation:
UCSD Department of Mathematics, 9500 Gilman Dr. #0112, La Jolla, CA 92093-0112, USA (email: drogalsk@math.ucsd.edu)
Susan J. Sierra
Affiliation:
School of Mathematics, James Clerk Maxwell Building, King’s Buildings, University of Edinburgh, Edinburgh EH9 3JZ, UK (email: s.sierra@ed.ac.uk)
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Abstract

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We construct an interesting family of connected graded domains of Gel’fand–Kirillov dimension 4, and show that the general member of this family is noetherian. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin–Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander–Buchsbaum formula also fails to hold for these algebras. The algebras we construct are birational to ℙ2, and their existence disproves a conjecture of the first author and Stafford. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on ℙ1×ℙ1, and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques.

Keywords

noncommutative projective geometrynoncommutative surfacesnoetherian graded ringsample sequence

MSC classification

Secondary: 16S38: Rings arising from non-commutative algebraic geometry 16P40: Noetherian rings and modules 16S80: Deformations of rings 16W50: Graded rings and modules

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1195 - 1237
Copyright
Copyright © Foundation Compositio Mathematica 2012

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