On Betti numbers for symmetric powers of modules

2024/02/14 by Jorge-Pérez, V. H., Lima, J. A.
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.09309

Abstract

Let M be a finitely generated module over a local ring (R,\mathfrakm). By S_j(M), we denote the jth symmetric power of M (jth graded component of the symmetric algebra S_R(M)). The purpose of this paper is to investigate the minimal free resolutions S_j(M) as R-module for each j≥ 2 and determine the Betti numbers of 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 S_j(M) in terms of Betti numbers of M. In particular, obtain some applications of the famous Buchsbaum-Eisenbud-Horrocks conjecture.

Related

• Total Betti numbers of modules of finite projective dimension
• The Regularity of Tor and Graded Betti Numbers
• Lower bounds on Betti numbers
• Betti numbers of Zⁿ-graded modules
• Polynomial growth of Betti sequences over local rings