1. Introduction

In computational algebraic geometry as well as in the theory of singularities it is a fundamental tool to compute free resolutions of ideals and modules. For example, for the computation of homology and cohomology groups, for Betti numbers and other invariants. Therefore, it is necessary to provide a computer algebra system with a fast implementation of syzygies.

In recent years several methods were developed to speed up computations of syzygies. Here I would like to give an overview of the essential ideas and our practical experiences with them.

Main contributions to that development come from W. Trinks, G. Zacharias, D.A. Spear, F. O. Schreyer, M. Stillman and D. Mumford, T. Mora and Moeller. Recent improvements to compute minimal resolutions directly have been created by C. Traverso, M. Caboara, R. La Scala and by myself.

In Chapter 2.2, as a new result, it is proved that the STZS-algorithm apply also to factor rings. Further, the technique of direct reductions is used to avoid Gaussian eliminations for computations in local orderings. It is described how detailed informations on Hilbert functions of the modules of a minimal resolutions can be obtained.

At the end suggestions for the algorithms for a global homogeneous local ordering are given.

I would like to thank G. Pfister for many fruitful discussions helping me to study this subject and the DFG for giving financial support during that time.