Embedding a quantum nonsingular quadric in a quantum P-3

[摘要] A definition of regularity has been given for non-commutative graded algebras and results of Artin, Schelter, Tate, and Van den Bergh classify the regular algebras of global dimension three that are generated by degree one elements. Our purpose is to classify a certain class of quadratic regular algebras of global dimension four. Let S be a twisted homogeneous coordinate ring of a nonsingular quadric Q subset of P-3. Our interest is in algebras R such that there is an embedding ProjS hooked right arrow Proj R. In this paper, we classify all the quadratic regular algebras R of global dimension four which have the same Hilbert series as that of the polynomial ring on four variables, and which map onto S via a graded degree zero homomorphism. Our approach makes use of the point modules of R and their associated geometric data. We classify the algebras R according to their ''point scheme'' P and corresponding automorphism sigma is an element of Aut(P); those algebras R which are determined by (P, sigma) belong to at most a five-parameter family, but those which are not determined by (P, sigma) belong to at most a four-parameter family. In the first case, P is either P-3 Or consists Of Q together with a line L, while in the second case P = Q. It is also proved that under certain sufficient conditions, the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism. (C) 1997 Academic Press.

[发布日期] 1997-09-01