Non-commutative Algebraic Geometry

0 Introduction This is a reasonably faithful account of the ve lectures I delivered at the summer course \Geometria Algebraica no Commutativa y Espacios Cuanti-cos" for graduate students, in Spain, July 25{29, 1994. The material covered was, for the most part, an abridged version of Artin and Zhang's paper 2]. Fix a eld k. Given a Z-graded k-algebra, A say, which for simplicity is assumed to be left noetherian and locally nite dimensional, its non-commutative projective scheme is deened to be the pair proj(A) := (tails(A); A); where tails(A) is the quotient category of grmod(A), the category of nitely generated graded left A-modules, modulo its full subcategory of nite dimensional modules, and A is the image of the distinguished module A A in tails(A). If A is a quotient of a commutative polynomial ring generated in degree 1, Serre 4] proved that proj(A) is isomorphic (in an obvious sense) to