We develop an algebraic theory of supports for \(R\)-linear codes fixed length, where \(R\) is a finite commutative unitary ring. A support naturally induces notion generalized weights and allows one to associate monomial ideal code. Our main result states that, under suitable assumptions, the code can be obtained from graded Betti numbers its associated ideal. In case \(\mathbb{F}_q\)-linear e...

We show that very weak topological assumptions are enough to ensure the existence of a Hellytype theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If F is a finite family of subsets of R such that β̃i ( ⋂G) ≤ b for any G ( F and every 0 ≤ i ≤ dd/2e−1 then F has Helly number at most h(b, d). Here β̃i denotes t...

A relatively compressed algebra with given socle degrees is an Artinian quotient A of a given graded algebra R/c, whose Hilbert function is maximal among such quotients with the given socle degrees. For us c is usually a “general” complete intersection and we usually require that A be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that f...

An important application of the algebraic theory of L2-Betti numbers [10] (see Farber [8] for an alternative approach) is that the L2-Betti numbers β i (Γ ) of a group Γ vanish if it has a normal subgroup whose L2-Betti numbers vanish. With regard to the first L2-Betti number, one can significantly relax the normality condition to obtain similar vanishing results [14]. Peterson and Thom prove i...

If M is the complement of a hyperplane arrangement, and A = H(M, k) is the cohomology ring of M over a field of characteristic 0, then the ranks, φk, of the lower central series quotients of π1(M) can be computed from the Betti numbers, bii = dimTor A i (k, k)i, of the linear strand in a minimal free resolution of k over A. We use the Cartan-Eilenberg change of rings spectral sequence to relate...

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded R-module, where R = k[x1, . . . , xm] is the polynomial ring over a field k in m variables. The bound is given in terms of the rank and the first two Betti numbers of the module. An example is given which achieves these bounds simultaneously in each homological degree. Using Alexander duality, a bound...

Extending results for space curves we establish bounds for the cohomology of a non-degenerate curve in projective n-space. As a consequence, for any given n we determine all possible pairs (d, g) where d is the degree and g is the (arithmetic) genus of the curve. Furthermore, we show that curves attaining our bounds always exist and describe properties of these extremal curves. In particular, w...

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley-Reisner ring R/IΛ and the inverse system algebra R/I∆. We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giv...

Schreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers b1 and b2, associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomia...