Computing Koszul Homology for Monomial Ideals

The Koszul homology of modules of the polynomial ring R is a central object in commutative algebra. It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we consider the case of modules of the form R/I where I is a monomial ideal. So far, some good algorithms have been given in the literature and implemented in different Computer Algebra Systems (e.g. CoCoa, Singular, Macaulay), which compute minimal free resolutions of modules of the form R/I with I an ideal in R, which include the case of I being a monomial ideal as a particular one (a good review is given in [10]). Our goal is to build algorithms specially targeted to monomial ideals, taking into account the particular combinatorial and structural properties of these ideals. This being a first goal, it is also a first step of an alternative approach to the computation of the Koszul homology and minimal free resolutions of polynomial ideals. 1 The Koszul Complex Let V be a vector space of dimension n and give it the basis {x1, . . . , xn} so that we can identify SV with R = k[x1, . . . , xn], and consider the complex K : 0 → R⊗ ∧V ∂ → R ⊗ ∧V ∂ → · · ·R⊗ ∧V ∂ → R⊗ ∧V → k → 0 where the maps are given by the rule ∂(w1 · · ·wq ⊗ xi0 ∧ · · · ∧ xil) 7−→ l ∑ j=0 (−1)w1 · · ·wqxij ⊗ xi0 ∧ · · · ∧ x̂il · · · ∧xil this is called the Koszul Complex, and it is a minimal free resolution of k. Given a graded module M, its Koszul Complex (K(M), ∂) is the tensor product complex M⊗R K. The Koszul homology of M is the homology of this product complex. We can identify this homology modules with Tor • (M,k), which can also be computed using any resolution of M and tensoring with k. ∗Partially supported by NEST Project 5006 (GIFT)