Monomial ideal

Abstract: A very powerful technique in commutative algebra was introduced by Macaulay, who realized that studying the initial terms of elements of an ideal gives one great insight into the algebra and combinatorics of the ideal. The initial ideal depends on the choice of coordinates, but there is an object, the initial ideal in generic coordinates, which is coordinate-independent. Generic initial ideals appeared in the work of Grauert and Hironaka.

Abstract: 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 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 · · ·xan n , a ∈ Z. Of special interest are the two cases when M has a minimal monomial generating set which is either finite or 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 ∈ 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 of M 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.

Abstract: For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

Abstract: Each partition A = (λ 1 , λ 2 ,..., An) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.