1. Introduction

The computation of syzygies and free resolutions of ideals and modules is one of the main applications of computer algebra. It is common folklore that a Groebner basis computation produces also all information about the module of syzygies of a set of generators. First implementations of free resolutions were based on this fact. In the last years several improvements of this approach were developped and implemented (see [5], [1] and [7]). The main ideas of these improvements are special reduction strategies based on F.-O. Schreyer´s monomial orderings and the usage of the Hilbert function to avoid useless pairs within the reduction process. All these algorithms use the traditional process of creating s-polynomials of pairs of polynomials and reducing them.

In abstract algebra there is another idea to compute free resolutions of ideals - the Koszul complex (see J.L. Koszul [4]). Its commutative variant is based on the concept of regular sequences of elements of a ring $R$ (see Chapter 17 of [3]).

In this paper we connect the ideas used in the Koszul complex with the computation of free resolutions based on Buchberger's algorithm. For this purpose the notion of a sequential algorithm is introduced in chapter 2. This kind of algorithm constructs the free resolution of an ideal via a sequence of subideals (which differ by one generator each time) and the extension of the associated free resolutions. Along this line we decide first the $I$-regularity of a new generator and use a tensor product of complexes in the positive case. The key-point of the algorithm is the handling of the non-$I$-regular generators. In Chapter 3.3 we construct a natural generalization of the Koszul complex for non-$I$-regular generators (which should not be confused with the generalized Koszul complex introduced by D. Buchsbaum [2]). This construction might give a deeper insight into the nature of free resolutions.

The algorithmic part of this new construction is mainly done by a special choise of orderings on the resolution which allows to put the different cases in a natural framework (see Chapter 4). Finally, the pseudo code of such an algorithm is presented and we give some timings based on an implementation within the computer algebra system SINGULAR.