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REES VALUATIONS FOR A FILTRATION WITH RESPECT TO A MODULE
Keywords:
valuation, filtration, module, ring, Krull domainDOI:
https://doi.org/10.17654/0972555525028Abstract
Let $R$ be a noetherian ring, $f=\left(I_n\right)_{n \in \mathrm{~N}}$ be a filtration on $R$ and $M$ be a finitely generated $R$-module. Consider the asymptotic function $\bar{v}_{f, M}(x)=\lim _{n \rightarrow+\infty} \frac{v_{f, M}\left(x^n\right)}{n}$, where $v_{f, M}(x)=\sup \{k \in \mathrm{~N} / x M \left.\subseteq I_k M\right\}, \quad x \in R$. Then we prove that under certain conditions on $f$, there exist finitely many discrete valuations that determine $\bar{v}_{f, M}(x)$.
Received: January 20, 2025
Revised: May 21, 2025
Accepted: July 14, 2025
References
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