We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.

We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certai...

Abstract. We use the results by Eisenbud and Schreyer [3] to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan [5] for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of th...

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue’s result: Let p ≥ 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a grade...

Let K be a field of characteristic 0 and S = K[x1, . . . , xn] the polynomial ring over K with graded maximal ideal m = (x1, . . . , xn). Denote by βi(M) = Tor S i (K,M) the ith Betti number of a finitely generated graded module M and by Gin(I) the generic initial ideal of a graded ideal I with respect to the reverse lexicographical order. In this paper we answer (positively) a question raised ...

Let R=k[x1,…,xn] denote the standard graded polynomial ring over a field k. We study certain classes of equigenerated monomial ideals with property that so-called complementary ideal has no linear relations on generators. then use iterated trimming complexes to deduce Betti numbers for such ideals. Furthermore, using result splitting mapping cones by Miller and Rahmati, we construct minimal fre...

Let I be an ideal generated by quadrics in a standard graded polynomial ring S over field. A question of Avramov, Conca, and Iyengar asks whether the Betti numbers R = / can bounded above binomial coefficients on minimal number generators if is Koszul. This has been answered affirmatively for Koszul algebras defined three almost complete intersections with any generators. We give strong affirma...

0. Introduction 1. L-Betti numbers for CW -complexes of finite type 2. Basic conjectures 3. Low-dimensional manifolds 4. Aspherical manifolds and amenability 5. Approximating L-Betti numbers by ordinary Betti numbers 6. L-Betti numbers and groups 7. Kähler hyperbolic manifolds 8. Novikov-Shubin invariants 9. L-torsion 10. Algebraic dimension theory of finite von Neumann algebras 11. The zero-in...

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in P, in particular, fat point schemes. We determine upper and lower bounds for the Hilbert function of A based on being given some (but not necessarily all and possibly none) of the maximal subsets of the collinear points of the support of A, i.e., given partial information about the matroid associat...