Cellular Resolutions of Monomial Modules

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals [BPS],[PS]. Introduction Given a field k, we consider the Laurent polynomial ring T = k[x 1 , . . . , x ±1 n ] as a module over the polynomial ring S = k[x1, . . . , xn]. The module structure comes from the natural inclusion of semigroup algebras S = k[N] ⊂ k[Z] = T . A monomial module is an S-submodule of T which is generated by monomials x = x1 1 · · · x an n , a ∈ Z . Every monomial moduleM has a unique minimal generating set of monomials. Of special interest are the two cases when this generating set is finite or when it forms a group under multiplication. In the first case M is isomorphic to a monomial ideal in S. In the second case M coincides with the lattice module ML := S {x a | a ∈ L} = k {x | b ∈ N + L} ⊂ T. for some sublattice L ⊂ Z whose intersection with N is the origin 0 = (0, . . . , 0). We shall derive free resolutions of M from regular cell complexes whose vertices are the generators ofM and whose faces are labeled by the least common multiples of their vertices. The basic theory of such cellular resolutions is developed in Section 1. Our main result is the construction of the hull resolution in Section 2. We rescale the exponents of the monomials in M , so that their convex hull in R is a polyhedron Pt whose bounded faces support a free resolution of M . This resolution is new and interesting even for monomial ideals. It need not be minimal, but, unlike minimal resolutions, it respects symmetry and is free from arbitrary choices. In Section 3 we relate the lattice module ML to the Z /L-graded lattice ideal IL = 〈 x − x | a − b ∈ L 〉 ⊂ S. This class of ideals includes ideals defining toric varieties. We express the cyclic Smodule S/IL as the quotient of the infinitely generated S-module ML by the action of L. In fact, we like to think of ML as the “universal cover” of IL. Many questions about IL can thus be reduced to questions about ML. In particular, we obtain the hull resolution of a lattice ideal IL by taking the hull resolution of ML modulo L. This paper is inspired by the work of Barany, Howe and Scarf [BHS] who introduced the polyhedron Pt in the context of integer programming. The hull resolution generalizes results in [BPS] for generic monomial ideals and in [PS] for generic lattice ideals. In these generic cases the hull resolution is minimal. We remark that deriving algebraic syzygies from polytopes is a natural operation also in other contexts such as associative algebras [AB] and algebraic K-theory [KS].