The Beilinson complex and canonical rings of irregular surfaces

In the first part of the paper Beilinson's theorem on the bounded derived category of coherent sheaves on P^n is extended to weighted projective spaces in a rather explicit form. To this purpose the usual category of coherent sheaves is replaced by a suitable category of graded sheaves, and a more general theory of graded schemes is developed. In the second part of the paper the weighted version of Beilinson's theorem is applied to prove a structure theorem for certain canonical projections of surfaces of general type into a 3-dimensional weighted projective space. This result (which generalizes to the weighted case a theorem by Catanese and Schreyer) is mainly interesting for irregular surfaces, and we illustrate it by studying a family of surfaces with p_g=q=2 and K^2=4, whose canonical rings are explicitly computed along the way.