Minimal resolutions

The methods of Effective Homology [5] give a simple algorithm computing the minimal resolution of an A0-module of finite type M0, when A0 is an ordinary polynomial ring A0 = k[x1, . . . , xm]0 localized at 0 ∈ k. Standard arguments allow us to study instead the global case of A = k[x1, . . . , xm], of an A-module M, and we are looking for an A0-resolution of M ⊗A A0. With respect to which seems the previously known methods [4, Section 4.8], the situation is the following. Our method is conceptually remarkably simple, once the very nature of effective homology is understood. On the contrary, the technicalities of the other methods are rather laborious, which of course does not mean useless. The style of our algorithm is quite different; effective homology can be seen as an automatic program writing process, deducing machine programs from simple notions of homological algebra, mainly the homological perturbation lemma. Experience in Algebraic Topology shows programs obtained in this way are simple, readable and efficient, the same in Commutative Algebra where other programs computing the effective homology of Koszul complexes have already shown the interest of the point of view and the efficiency of the programs that are so obtained.