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 (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 -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--regular generators. In Chapter 3.3 we construct a natural generalization of the Koszul complex for non--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**.