The asymptotic Samuel function of a filtration (Q6563078)

There is a rich literature about the asymptotic Samuel function of ideals of commutative Noetherian rings. In this paper, the authors define the notion of asymptotic Samuel function for filtrations of ideals. Let \(R\) be a commutative Noetherian ring with identity and \(\mathcal{I}=\{I_m\}_{m\in \mathbb{N}}\) be a filtration of ideals of \(R\). They define the asymptotic Samuel function of \(\mathcal{I}\) as the function \N\[ \begin{cases} \N\overline{\nu}_{\mathcal{I}}: R\rightarrow \mathbb{R}_{\geq 0}\cup \{\infty\} \\\N\overline{\nu}_{\mathcal{I}}(x)=\underset{n\to\infty} \lim \frac{\nu_{\mathcal{I}}(x^n)}{n}, \N\end{cases} \] \Nwhere \(\nu_{\mathcal{I}}(f)=\sup \{m \mid f\in I_m\}\) for every \(f\in R\). If \(I\) is an ideal of \(R\) and \(\mathcal{I}\) is the \(I\)-adic filtration of \(I\), then \(\overline{\nu}_{\mathcal{I}}\) is equal to the classical asymptotic Samuel function \(\overline{\nu}_{I}\) of the ideal \(I\).\N\NThe authors show that many of the good properties of the asymptotic Samuel function of an ideal of \(R\) remain true for the asymptotic Samuel function of a general filtration of ideals of \(R\). In particular, they extend the notion of projectively equivalent ideals to filtrations. Recall that two ideals \(I\) and \(J\) of \(R\) are said to be projectively equivalent if \(\overline{\nu}_{I}=\alpha\overline{\nu}_{J}\) for some \(\alpha\in \mathbb{R}_{>0}\). The authors call two filtrations of ideals \(\mathcal{I}\) and \(\mathcal{J}\) projectively equivalent if \(\overline{\nu}_{\mathcal{I}}=\alpha\overline{\nu}_{\mathcal{J}}\) for some \(\alpha\in \mathbb{R}_{>0}\). It is known that if two ideals \(I\) and \(J\) and \(\alpha\in \mathbb{R}_{>0}\) are such that \(\overline{\nu}_{I}=\alpha\overline{\nu}_{J}\), then \(\alpha\) is rational. Although, the authors show that this does not necessarily remain true for general filtrations of ideals, they prove the following nice characterization of projectively equivalent filtrations:\N\N{Theorem:} Let \(\mathcal{I}=\{I_m\}_{m\in \mathbb{N}}\) and \(\mathcal{J}=\{J_m\}_{m\in \mathbb{N}}\) be two filtrations of ideals of \(R\). Then \(\mathcal{I}\) and \(\mathcal{J}\) are projectively equivalent if and only if there are \(\alpha, \beta\in \mathbb{R}_{>0}\) such that the integral closures of \(R[\mathcal{I}^{(\alpha)}]\) and \(R[\mathcal{I}^{(\beta)}]\) in \(R[t]\) are the same.\N\N(Here, for any filteration \(\mathcal{I}=\{I_m\}_{m\in \mathbb{N}}\) of ideals of \(R\) and any \(\alpha\in \mathbb{R}_{>0}\), the fiteration \(\mathcal{I}^{(\alpha)}\) is defined by \(\mathcal{I}^{(\alpha)}:= \{I_{[\alpha m]}\}_{m\in \mathbb{N}}\). Also, \(R[\mathcal{I}]=\sum \limits_{i\in \mathbb{N}}I_mt^m\subseteq R[t]\) denotes the Rees algebra of a filteration \(\mathcal{I}=\{I_m\}_{m\in \mathbb{N}}\).)