Some noncommutative projective surfaces of GK-dimension 4

Details Some noncommutative projective surfaces of GK-dimension 4 Some noncommutative projective surfaces of GK-dimension 4

2011-01-04

arxiv.org/abs/1101.0737v2

Mathematics

Rings and Algebras

48 pages, 1 figure, comments welcome; v2 introduction rewritten, no other significant changes

Scientific paper

We construct a family of connected graded domains of GK-dimension 4 that are birational to P2, and show that the general member of this family is noetherian. This disproves a conjecture of the first author and Stafford. 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 our family. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on P1xP1, 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.

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