Taylor and minimal resolutions of homogeneous polynomial ideals

In the theory of monomial ideals of a polynomial ring S over a field k, it is convenient that for each such ideal I there is a standard free resolution, so called Taylor resolution, that can be canonically constructed from the minimal system of monomial generators of I (see [7], p.439 and section 2). On the other hand no construction of a minimal resolution for an arbitrary monomial ideal has been known. Recently a minimal resolution was constructed in [4] for a class of so called generic monomial ideals. Also in [2, 10, 11] various invariants of monomial ideals were related to combinatorics of the lattice D of the least common multiples (lcm) of generating monomials. In particular in [11] the Betti numbers of the S-module S/I were expressed through homology of D and it was proved that even the algebra structure of TorS∗ (S/I, k) was defined by that lattice although explicit formula was not given in that paper. Given a system of generators of an arbitrary ideal I of S, one can factor the generators in irreducibles and construct the Taylor complex similarly to the Taylor resolution of a monomial ideals. In general this complex is not acyclic. One non-monomial case where it is acyclic was used in [15]. In the present paper, a necessary and sufficient condition is given for the Taylor complex of a system A of homogeneous polynomials to be acyclic (Theorem 2.4). This condition involves the local homology of the lattice D of lcm of elements form A and the depth of ideals generated by their irreducible factors. If this condition holds then the Betti numbers of S/I are defined by the local homology of D similarly to the case of monomial ideals (Theorem 2.5). Moreover in section 3 we exhibit a DGA defined by combinatorics whose cohomology algebra is isomorphic to the algebra TorS∗ (S/I, k). This construction makes sense for arbitrary graded lattice (see definition in section 3) and generalizes the DGA constructed in [16, 17] for cohomology algebra of complex subspace complement. Section 4 contains that can be considered the main result of the paper (Theorem 4.3). There for any ideal I having the Taylor resolution we give a combinatorial construction of a subcomplex of it that is a minimal resolution of S/I. This construction is not canonical and involves computations of homology of posets which hardly can be avoided in general. We also describe completely the class of ideals for which our minimal resolution reduces to the minimal resolution from [4]. Finally in section 5 we give examples of classes of A satisfying the