Strategies for Computing Minimal Free Resolutions

One of the most important computations in algebraic geometry or commutative algebra that a computer algebra system should provide is the computation of finite free resolutions of ideals and modules. Resolutions are used as an aid to understand the subtle nature of modules and are also a basis of further computations, such as computing sheaf cohomology, local cohomology, Ext, Tor, etc. Modern methods for calculating free resolutions derive from the theory of Gröbner bases. These methods were introduced at the end of the 1970s by Richman (1974); Spear (1977); Schreyer (1980) and have survived in computer algebra systems up to now. However, the problem with these algorithms is that many computations of interest for researchers were out of range. This is giving impulse to authors such as Capani et al. (1997), Siebert (1996) and ourselves to develop decisive improvements of the resolution techniques. Resolution algorithms based on Gröbner bases can be divided essentially into two types. The first type is based on computing the syzygy module on a minimal set of generators. The second type, initially used by Frank Schreyer, is based on computing the syzygy module on a Gröbner basis. In both cases, using induced term orderings leads to a large improvement in the sizes of the Gröbner bases involved. Which of these two methods is best depends in part on the specific input ideal or module. However, we have found that for problems of interest, the Schreyer technique, together with the improvements that we suggest, on the average outperforms the other methods.