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...

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Bibliographic Details
Published inarXiv.org
Main Authors Jorge-Pérez, V H, Lima, J A
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.02.2024
Subjects
Lower bounds
Modules
Rings (mathematics)
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Summary: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.
ISSN:2331-8422