On Betti numbers for symmetric powers of modules and some applications - Lima, Johnny Albert dos Santos (2022)
Let M 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 thesis 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 about the famous Buchsbaum-Eisenbud-Horrocks conjecture.