On Betti numbers for symmetric powers of modules

Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m})$. By ${\mathcal{S}}_j(M)$, we denote the $j$th symmetric power of $M$ ($j$th graded component of the symmetric algebra ${\mathcal{S}}_R(M)$). The purpose of this paper is to investigate the minimal free resolutions ${\mathcal{S}}_j(M)$ as $R$-module for each $j\ge 2$ and determine the Betti numbers of ${\mathcal{S}}_j(M)$ 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 $I^j$ in terms of the Betti numbers of $I$. In addition, we establish upper and lower bounds of Betti numbers of ${\mathcal{S}}_j(M)$ in terms of Betti numbers of $M$. In particular, obtain some applications of the famous Buchsbaum-Eisenbud-Horrocks conjecture.