Computation of the (n-1)-st Koszul Homology of monomialideals and related algorithms

Abstract

Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n - 1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions. This feature of (n - 1)-st Koszul homology allows us to transform an algorithm that computes Koszul homology of monomial ideals to use it for the computation of irreducible and Stanley decompositions. This is an example of how algorithms and structures specifically targeted to computations on monomial ideals should take into account the combinatorial properties of them to produce efficient methods, an issue that is worth introducing into modern computer algebra systems. To illustrate this fact we present some details on the implementation of the algorithm in CoCoALib.