Homological description of Monomial Ideals using Mayer-Vietoris Trees

The goal of this paper is to provide homological descriptions of monomial ideals. The key concepts in these descriptions are the minimal free resolution, the Koszul homology and the multigraded Betti numbers. These three objects are strongly related, being Tor modules a simple way to describe this relation. We introduce a new tool, Mayer-Vietoris trees which provides a good way to compute the homological description of monomial ideals. They can be used to compute either minimal free resolutions, Koszul homology or multigraded Betti numbers of monomial ideals. Related with Mayer-Vietoris trees, we introduce the families of Mayer-Vietoris ideals, which include several well known families of monomial ideals as particular cases. Several examples and applications are also provided. Introduction Let R = k[x1, . . . , xn] the ring of polynomials in n variables over a field k of characteristic 0, and I ⊆ R a monomial ideal. Many homological invariants and properties of I can be read from the (multigraded) Betti numbers of it. These include, depth, dimension, Castelnuovo-Mumford regularity, etc. Being multigraded Betti numbers the ranks of the modules in a minimal resolution of I, they can be considered as the ranks of the multigraded Tor(I,k) modules, and also as the ranks of the Koszul homology modules of I. These equivalences origin from equivalent ways to compute Tor(I,k). Definition 0.1. Let M be an R-module. The n-th left derived functor of the right-exact functor M ⊗− is denoted by Tor n (M,−). Following the definition of left derived functor, the computation of Tor • (k, I) goes as follows: Take a resolution P of I as an R-module, and tensor it with k. Then, Tor i (I,k) is just the i-th homology module of the tensor complex P ⊗ k. This homology is independent of the resolution taken. If P is a minimal resolution of I, then tensoring it with k yields a complex with zero differentials in every dimension, and the ranks of the homology modules are just the ranks of the modules in P (see [2] for example) i.e. the Betti numbers of I. Monomial ideals being multigraded modules of R, their minimal free resolutions are also multigraded and hence their Betti numbers and Tor modules. On the other hand, one can use a resolution of k as R-module, and tensor it with I, the homology of this product complex is again Tor(I,k). The Koszul complex, let us denote it K, provides a resolution of k [4], and I ⊗ K is by definition the Koszul complex of I, which is no longer a resolution; the homology of this complex is called the Koszul homology of I. In this paper we use the following definition of Koszul homology: Let V be a n-dimensional k-vector space. Let SV and ∧V be the symmetric and exterior algebras of V respectively. We consider the basis of V given by {x1, . . . , xn}; then we can identify SV and R and consider the following complex K : 0→ R⊗ ∧V ∂ → R⊗ ∧n−1V ∂ → · · ·R⊗ ∧V ∂ → R⊗ ∧V → k→ 0 ∗Partially supported by NEST-Adventure contract 5006 (GIFT) and project ANGI 2005/10