On the Hilbert Coefficients and Betti Numbers of the Stanley-Reisner Ring of a Matroid Complex

Let S = K[xi,...,x] be a polynomial ring. Herzog and Zheng conjectured that the i-th Hilbert coefficient of a finitely generated graded Cohen-Macaulay S-module N generated in degree 0 is bounded by the functions of the minimal and maximal shifts in the minimal graded free resolution of N over S and the 0-th Betti number of N. Also, Romer asked whether under the Cohen-Macaulay assumption the i-th Betti number of S/I, where I c S is a graded ideal, can be bounded by the functions of the minimal and maximal shifts of S/I. In this paper, we provide elementary proofs to establish Herzog and Zheng's conjecture and the upper bound part of Romer's question for the Stanley-Reisner ring of a matroid complex. 2010 Mathematics Subject Classification: 13D99, 13D40, 05B35