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arXiv · 1409.0816

Growth of multiplicities of graded families of ideals

Huy Tai Ha·Pham An Vinh
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Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $\gamma > 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < \gamma n^{d-1}.$$

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Huy Tai Ha, Pham An Vinh. 2014-09-02. Growth of multiplicities of graded families of ideals. https://arxiv.org/abs/1409.0816

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