Kustin-Miller unprojection with complexes

A main ingredient for the Kustin-Miller unprojection is the module Hom(R)(I, omega(R)), where R is a local Gorenstein ring and I a codimension one ideal with R/1 Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of Catanese, Franciosi, Hulek, and Reid (Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999), 185220). The second and the third are about Tom and Jerry, two families of Gorenstein codimension four rings with 9 x 16 resolutions