Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: • constructing the minimal free resolutions of special monomial ideals, cf. [AHH, BPS] • constructing non-minimal free resolutions; for example, Taylor’s resolution (cf. [Ei, p. 439]) and the cellular resolutions •...

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, . . ...

Let A be an ample line bundle on an abelian variety X (over an algebraically closed field). A theorem of Koizumi ([Ko], [S]), developing Mumford’s ideas and results ([M1]), states that if m ≥ 3 the line bundle L = A⊗m embeds X in projective space as a projectively normal variety. Moreover, a celebrated theorem of Mumford ([M2]), slightly refined by Kempf ([K4]), asserts that the homogeneous ide...

We explore the structure of non-redundant and minimal sets consisting of graded if-then rules. The rules serve as graded attribute implications in object-attribute incidence data and as similarity-based functional dependencies in a similarity-based generalization of the relational model of data. Based on our observations, we derive a polynomial-time algorithm which transforms a given finite set...

Let k be a field, and let R = k[x1, x2, x3]. Given a Hilbert function H for a cyclic module over R, we give an algorithm to produce a stable ideal I such that R/I has Hilbert function H and uniquely minimal graded Betti numbers among all R/J with the same Hilbert function, where J is another stable ideal in R. We also show that such an algorithm is impossible in more variables and disprove a re...