Let A = {a1, . . . , am} ⊂ Z be a vector configuration and IA ⊂ K[x1, . . . , xm] its corresponding toric ideal. 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. We associate to A a simplicial complex ∆ind(A). We show that the… (More)

Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a monomial ideal of S. It is well known that the Poincaré series of k over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.

Let A be a semigroup whose only invertible element is 0. For an A-homogeneous ideal we discuss the notions of simple i-syzygies and simple minimal free resolutions of R/I. When I is a lattice ideal, the simple 0-syzygies of R/I are the binomials in I. We show that for an appropriate choice of bases every A-homogeneous minimal free resolution of R/I is… (More)

Let R = k[x1, . . . , xn] be a polynomial ring over a field k. We present a characterization of multigraded R-modules in terms of the minors of their presentation matrix. We describe explicitly the second syzygies of any multigraded R-module.

Let k be a field, L ⊂ Zn be a lattice such that L∩Nn = {0}, and IL ⊂ k[x1, . . . , xn] the corresponding lattice ideal. We present the generalized Scarf complex of IL and show that it is indispensable in the sense that it is contained in every minimal free resolution of R/IL.

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. In the second part we associate to A a simplicial complex ∆ind(A). We show that the vertices of… (More)