Hara Charalambous

MINIMAL SYSTEMS OF BINOMIAL GENERATORS AND THE INDISPENSABLE COMPLEX OF A TORIC IDEAL

Let A = {a1, . . . , am} � Zn be a vector configuration and IA � K(x1, . . . , xm) its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of IA. We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to A a simplicial complexind(A). We show that the vertices ofind(A) correspond to the indispensable monomials of the toric ideal IA, while one dimensional facets ofind(A) with minimal binomial A-degree correspond to the indispensable binomials of IA.

Betti numbers of multigraded modules

In this paper we deal with lower bounds for the Betti numbers of multigaded modules over R = k[x1,…, xd]. In general the ith Betti number of a finite length multigraded module must be at least the binomial coefficient (di). This is achieved only when the module in question is isomorphic to R modulo a maximal R-sequence. Otherwise the lower bound for each i is increased either by (d−1i) or by (d−1i−1). If M is an arbitrary multigraded module and s is the length of a maximal R sequence in the annihilator of M then similar inequalities hold for the Betti numbers of M with d replaced by d−s.

Poincare series and resolutions of the residue field over monomial rings

Let, S = k[x1, …, xn] be a. polynomial ring over a field k, I a monomial ideal of S, R - S/I. When the D. Taylor resolution of R over S is minimal we construct a resolution of k over R . We discuss the corresponding multigraded Poincare series of R and expand on Frobergs formula

Minimal generating sets of lattice ideals

Let

Binomial fibers and indispensable binomials

L\subset \mathbb {Z}^n

On the Denominator of the Poincaré Series for Monomial Quotient Rings

L⊂Zn be a lattice and

Topological constructions for multigraded squarefree modules

I_L=\langle x^{\mathbf {u}}-x^{\mathbf {v}}:\ {\mathbf {u}}-{\mathbf {v}}\in L\rangle