Syzygies play an important role in computational algebraic geometry. They form a cornerstone for basic algorithms and constructions. For example, the computation of coherent sheaf cohomology on projective varieties is based on syzygy computations over exterior algebras via the theory of Tate resolutions. In this project we plan to advance the concept of syzygies further, both from a theoretical and from a computer algebra point of view. Goals are an algorithm to compute higher direct image complexes, improvement and parallelization of the basic syzygy algorithm and the investigation of asymptotics of syzygies.