Curves in Grothendieck Categories

Noncommutative projective geometry studies noncommutative graded rings by replacing the variety by a suitable Grothendieck category. One way of studying the resulting category is to examine the full subcategories which behave like curves on a commutative variety. Smith and Zhang initiated such a study by considering the subcategory generated by a particular type of module they called a pure curve module in good position. This paper generalizes their construction by allowing more general modules. The resulting category is shown to be categorically equivalent to a quotient of the category of graded modules over a graded ring. In the course of defining the category equivalence, several dimensions, including projective, injective and Krull dimensions, are calculated. In particular, this extension allows examination of the category created from a line module over more general AS-regular rings than those considered by Smith and Zhang. For instance, suppose that C is a generic line module over R_d, Stafford's Sklyanin-like algebra. Let C denote the category C generates. Then C is equivalent to the category of graded k[x, y]/(x² – y²) modules under the Z × Z/2Z-grading where deg(x) = (–1, 0) and deg(y) = (–1,1).