Generalizations of graded Clifford algebras and of complete intersections

For decades, the study of graded Clifford algebras has provided a theory where commutative algebraic geometry has dictated the algebraic and homological behavior of a noncommutative algebra. In particular, it is well known that a graded Clifford algebra, C, is quadratic and regular if and only if a certain quadric system associated to C is base point free. In this article, we introduce a generalization of a graded Clifford algebra, namely a graded skew Clifford algebra, and to such an algebra we associate a notion of quadric system in the spirit of the noncommutative algebraic geometry of Artin, Tate and Van den Bergh. We prove that a graded skew Clifford algebra is quadratic and regular if and only if its associated quadric system is normalizing and base point free. To prove our results, we extend the notions of complete intersection, base point free, quadratic form and symmetric matrix to the noncommutative setting. We use our results to produce several families of quadratic regular skew Clifford algebras of global dimension four that are not twists of graded Clifford algebras. Many of our examples have a 1-dimensional line scheme and a point scheme that consists of exactly twenty distinct points, and thus are candidates for generic regular algebras of global dimension four.