A formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.

Given multigraded free resolutions of two monomial ideals we construct a multigraded free resolution of the sum of the two ideals. Introduction Let K be a field, S = K[x1, . . . , xn] the polynomial ring in n variables over K, and let I and J be two monomial ideals in S. Suppose that F is a multigraded free S-resolution of S/I and G a multigraded free S-resolution of S/J . In this note we const...

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

(Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals J with pd(R/J) ≤ μ(J) − 1, i.e. whose projective dimension equals the minimal number of generators of J minus 1. In this paper we prove sufficient and necessary combinatorial conditions for pd(R/J) ≤ μ(J)−2. The second main result is an effective explicit procedure to compute the projective...

Let S = C[x1, . . . , xn+1] be the set of polynomials in n+1 variables over C (usually corresponding to the homogeneous coordinate ring of P). Let ≻ be the reverse lexicographical order on the monomials of S. Given a homogeneous ideal I in S, gin(I), the generic initial ideal of I, is the Borel-fixed monomial ideal associated to I. (For details see [B], [BM], [BS] or [Gr].) The generic initial ...

In this thesis, we undertake a combinatorial study of certain aspects of triangulations of cartesian products of simplices, particularly in relation to their relevance in toric algebra and to their underlying product structure. The first chapter reports joint work with Samu Potka. The object of study is a class of homogeneous toric ideals called cut ideals of graphs, that were introduced by Stu...

We give a new method to construct minimal free resolutions of all monomial ideals. This relies on two concepts: one is the well-known lcm-lattice ideal; other concept called Taylor basis, which describes how resolution can be embedded in resolution. An approximation formula for ideals also obtained.

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame. Introduction Let R be a graded ring. Recall that a graded R-module N is tame, if there exists an integer j0 such that Nj = 0 for all j ≤ j0, or else Nj 6= 0 for all j ≤ j0. Brodmann and Hellus [4] raised the question whether for a fin...

Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math, calling for the decomposition of ideals with a large amount of minimal generators. This paper presents an algorithm to compute such decompositions along with benchmarks showing a performance improvement by a factor of up to more than 1000. Performance can be further improved by tailo...