On Betti numbers for symmetric powers of modules
V. H. Jorge-Pérez, J. A. Lima
TL;DR
The paper investigates how the Betti numbers of symmetric powers $\mathcal{S}_j(M)$ of a finitely generated module $M$ over a Noetherian local ring relate to the Betti numbers of $M$, introducing the $(SW_j)$ condition to ensure finite free resolutions of $\mathcal{S}_j(M)$. A central construction, the complex $\mathcal{S}_j\mathbf{F}_{\bullet}$ built from a free resolution $\mathbf{F}_{\bullet}$ of $M$, yields exactness criteria via grade conditions on minors of the maps involved, and under these conditions provides minimal free resolutions of $\mathcal{S}_j(M)$ with explicit projective-dimension behavior. The authors derive explicit formulas for the Betti numbers $\beta_t^R(\mathcal{S}_j(M))$ in terms of the Betti numbers of $M$, giving special cases and corollaries for linear-type modules and their Rees algebras, including for powers $I^j$ of ideals. They establish general upper and lower bounds for these Betti numbers and connect the results to BEH and TR conjectures, presenting several PD-1 scenarios (and fiber-product contexts) where BEH-type lower bounds hold for $\mathcal{S}_j(M)$ and $I^j$, thereby offering new evidence and applications of these classical conjectures in the setting of symmetric powers.
Abstract
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 $j\geq 2$ 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.