Hilbert polynomial of length functions
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866911927977902080 |
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| author | Fornasiero, Antongiulio |
| author_facet | Fornasiero, Antongiulio |
| contents | Let $λ$ be a general length function for modules over a Noetherian ring R. We use $λ$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$λ$. We show that the leading term $μ$ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for $R[X]$-modules.
Similar to algebraic entropy, $μ$ in general is not additive for exact sequence of $R[X]$-modules: we demonstrate how to adapt of certain entropy constructions to this new invariant.
We also consider multi-variate versions of the Hilbert polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_04917 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hilbert polynomial of length functions Fornasiero, Antongiulio Commutative Algebra 13D40, 16S50 (Primary) 16D10, 16P40 (Secondary) Let $λ$ be a general length function for modules over a Noetherian ring R. We use $λ$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$λ$. We show that the leading term $μ$ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for $R[X]$-modules. Similar to algebraic entropy, $μ$ in general is not additive for exact sequence of $R[X]$-modules: we demonstrate how to adapt of certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial. |
| title | Hilbert polynomial of length functions |
| topic | Commutative Algebra 13D40, 16S50 (Primary) 16D10, 16P40 (Secondary) |
| url | https://arxiv.org/abs/2102.04917 |