In this chapter we develop the basic technique for calculating syzygies. It applies to the subvarieties Y in an affine space X with a desingularization Z which is a total space of a vector bundle over some projective variety V, which is a subbundle of the trivial bundle X × V over V. In such situation the Koszul complex of sheaves on X × V resolving the structure sheaf of Z has terms that are pullbacks of vector bundles over V. Taking the direct image of this Koszul complex by the projection p : X × V → V, one gets the formula expressing terms on the free resolution of the coordinate ring of Y in terms of cohomology of bundles on V. One also gets interesting complexes by taking direct images of the Koszul complex twisted by a pullback of a vector bundle on V. In this chapter we discuss the general construction and properties of direct images of Koszul complexes. The examples will be given in following chapters.

The chapter is organized as follows. In section 5.1 we state the properties of the twisted direct images F(V)• of Koszul complexes. In particular we give the expressions for their terms and homology.