On Betti numbers for symmetric powers of modules

Let M be a finitely generated module over a local ring (R, m). By Sⱼ (M), we denote the jth symmetric power of M (jth graded component of the symmetric algebra SR (M) ). The purpose of this paper is to investigate the minimal free resolutions Sⱼ (M) as R-module for each j 2 and determine the Betti numbers of Sⱼ (M) in terms of the Betti numbers of M. This has some applications, for example for linear type ideals I, we obtain formulas of the Betti numbers Iʲ in terms of the Betti numbers of I. In addition, we establish upper and lower bounds of Betti numbers of Sⱼ (M) in terms of Betti numbers of M. In particular, obtain some applications of the famous Buchsbaum-Eisenbud-Horrocks conjecture.